Talk:Integral/Archive 3

Cleaning up the lead
I've tried to clean up the remainder of the lead, focusing on making things tighter. I feel things like examples should be deferred to the article itself (an informal discussion section at the beginning for example) where they can be properly fleshed out. An effort on that front can be found at the Integral (sandbox version) page; any help there would be appreciated. Still missing from the lede are: As always, any help in adding that succinctly and elegantly to the lede is most welcome. -- Leland McInnes 16:14, 6 June 2007 (UTC)
 * Discussion/mention of differing definitions of integral (Riemann/Lebesgue, etc.)
 * Mention of fields of application.
 * Mention of techniques for computing integrals.


 * Notice that "lede" is not a word. The word is "lead" as in Lead section. JRSpriggs 10:01, 10 June 2007 (UTC)


 * It is actually: see News_writing. It is a traditional (archaic) spelling of "lead" originally used in journalism and printing to avoid confusion with the lead type of old printing presses. Quite a few Wikipedians use it. Like you, though, I think the conventional spelling is more appropriate in Wikipedia. Geometry guy 10:34, 10 June 2007 (UTC)

Informal discussion
I have added an informal discussion to try and provide some more accessible descriptions of integrals than the formal definitions. What currently stands is a first cut, and is perhaps a little long. It could also benefit from a few diagrams. Any and all assistance is welcome. -- Leland McInnes 19:54, 12 June 2007 (UTC)


 * There should definitely be an informal discussion part, but the example used is unfortunate:"The second difficulty, linked with the first, is there are not finitely many measurements of instantaneous speed". Measurements are in their nature always finite. We might make a lot of meauserements but it will always be a finite number. However, there is nothing wrong with the general idea. JKBlock 16:20, 18 June 2007 (UTC)


 * True, but I was thinking of the measurements as being "given" as by an oracle of some kind, rather than physically taken. The alternative is to suggest approximating a finite set of measurements by a function over a continuous domain, but that only complicates the issue. I might try and add soem clarification on this point though. -- Leland McInnes 14:50, 19 June 2007 (UTC)

Introduction
Why not introduce what the actual resut of an integral is in the introduction. Surely the whole point of an article is to sumarise, then explain. I dont see how just having what an actual integral looks like without its result is useful in the introduction.


 * If you are the anon who added the $$ = F(b) - F(a)$$, and the sentence "F(a) signifies the integral at the upper limit, similarly for F(b)", then the issue is that this is "the result" of an integral only through the Fundamental Theorem of Calculus, and not, for example, the result according to any of the standard formal definitions (which would involve limits or supremums). Introducing this as "the integral" serves to introduce a degree of confusion as to what an integral actually is. Also, the sentence added doesn't appear to be correct as stated, according to the definition of integral (that is, the definite integral, as opposed to the anti-derivative) used in the article. I agree that some mention of the FtoC in the intro may not go astray, but I suspect it would be better couched in a summary of the history of integrals. -- Leland McInnes 15:08, 13 June 2007 (UTC)

Numerical quadrature
Mathematicians tend to concentrate on the calculus/analysis aspects, so I dodged the competition by rewriting the section on numerical approximation. References forthcoming. Kahaner, Moler, &amp; Nash is one; Stoer &amp; Bulirsch is another. Maybe also the new Dahlquist &amp; Björck, which is currently available online. Suggestions welcome. Enjoy. --KSmrqT 23:21, 14 June 2007 (UTC)


 * Simpson's rule would be a good addition to the picture, but the numerical approximation section is already pretty long, so it is up to you. Cheers.--Cronholm144 23:50, 14 June 2007 (UTC)


 * As you noticed, one of the challenges was to say just enough, leaving a thorough study to the dedicated article. Simpson's rule adds nothing; technically, two Romberg steps are equivalent. If I had to reduce the section to one thought, it is that the calculus-book idea of a rectangle rule is practically worthless; serious modern algorithms typically use some form of adaptive Gaussian quadrature, which is vastly more accurate and efficient. Maybe eventually I'll find a way to make this vital point more briefly and more clearly, but the pre-rewrite version did not make the point at all! --KSmrqT 02:29, 15 June 2007 (UTC)

Lebesgue vs Riemann image is correct
Hi KSmrq, I understand why you would have doubt about this image because it's not clear that using simple functions corresponds to using horizontal slices, but that is really what's going on. Loisel 02:45, 17 June 2007 (UTC)


 * Sorry, no; it's not helpful. I refer you, for example, to this discussion (already alluded to in my edit summary). Please do not restore the image again without discussion. To that end, I'll ask WT:WPM for a broader spread of views, in case you and I alone are too close to see clearly. Thanks in advance for your patience. --KSmrqT 17:23, 17 June 2007 (UTC)


 * Well that is certainly how I teach the Lebesgue integral in the real analysis class. So you've got a contingent of mathematicians who think that this is the correct picture. Loisel 18:41, 17 June 2007 (UTC)


 * Though I've no knowledge of Lebesgue integrals, I agree with Loisel that horizontal slices is how most textbooks represent the Lebesgue integral. Loom91 08:40, 18 June 2007 (UTC)


 * Really? Which book? I have 3 Real Analysis books, and I have never seen the horizontal slice picture in any of them. –King Bee (&tau; • &gamma;) 10:29, 18 June 2007 (UTC)


 * Many books wouldn't have a "discussion" because of the level of the material, but one of the standard books is Folland's Real Analysis, and it does have a discussion. To quote from my edition, page 56: "In particular, if one picks the sequence constructed in the proof of Theorem (2.10a), one is in effect partitioning the range of f into subintervals Ij..." Perhaps we can add this quotation and a note (see Fig. (XXX)) and the figure that was deleted. Loisel 21:02, 18 June 2007 (UTC)
 * Ask if you need help adding references; we're using Harvard, not footnotes. (The numerous instances already in the article demonstrate the necessary incantations.) --KSmrqT 22:16, 18 June 2007 (UTC)

(unindent) No, KSmrq, you are using Harvard rather than footnotes. The article can be developed this way if you wish, but you must not revert other editors contributions because they do not conform to this style. If you wish to impose this style on the article (on the acceptable grounds that you were the initiating editor), you should edit contributions which do not conform, not delete them. This is simply courtesy, and I believe you are a courteous editor. Geometry guy 23:03, 18 June 2007 (UTC)
 * Are you trying to pick a fight?! We are using Harvard style, and I intend to be a pest about it. I'm perfectly happy to help Loisel, as I indicated in both my edit summary and here, should I be asked. At the moment the article has a list of 11 citations listed under "References", and I spent quite some time researching and formatting them, so the burden of courtesy lies on those stragglers adding a new reference.
 * I notice that the "footnote" in History was introduced when text was imported from the sandbox version. That was a mistake, of course; and the inattention to form is accompanied by an unacceptably sub-par citation. (Have you looked at the web page cited?) So I just deleted the whole thing. If someone wants to add substantial history references, I'd be delighted. I might even offer to help!
 * Meanwhile, I noticed you were kind enough to do some of the formatting for Loisel; so I went to the trouble to finish what the two of you started. I found and added the publisher, the edition, and the ISBN for Folland. (There is a second edition dated April 7, 1999 (ISBN 978-0-471-31716-6), but since the contents are not viewable online I did not feel at liberty to alter the citation.) And I moved the page number to the referencing context, which is where this one belongs. Since I am a courteous editor. --KSmrqT 02:01, 19 June 2007 (UTC)


 * You are both courteous and skilled editors... there seems to have been somewhat of a misunderstanding of intent, but we are all working towards the same goal, Right? --Cronholm144 02:28, 19 June 2007 (UTC)


 * Indeed, and thanks! This was just a friendly chastisement, because I know KSmrq has high standards, and indeed his response was admirable. It was a pity I missed the chance to join in the drinks and fine sentiments: mine would have been a long cold beer ;) Geometry guy 11:11, 19 June 2007 (UTC)

Improper integral
The statement beneath the improper integral: "An "improper" integral; when x = a is a point where f(x) becomes infinite." is argueable. f(x) does not 'become infinity'; the improper integral (in this version) is used when the function is not defined at x = a, and that's "why" the limit is used.  Also I think the notation  $$\int_{a}^{b} f(x)dx = \lim_{c \to a} \int_{c}^{b} f(x)dx$$  (with a note that a is either a real number or infinity) is better since it covers both integrals where the function isn't defined at a point and integrals on unbounded intervals. Please comment. I'm looking forward to joining the mathematics project on wikipedia :) JKBlock 20:00, 17 June 2007 (UTC)
 * Yeah, a good suggestion.. i just added it in quickly before i went back to work, good idea, though. :-) EDIT: Someone changed it since i added it.. is there an edit war or conflict here? *sigh* ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ slurp me! 16:25, 16 July 2007 (UTC)
 * A very good suggestion. Feel free to implement it. Loom91 08:39, 18 June 2007 (UTC)


 * I've changed some of the others things on this subject as well (for instance it also said that for proper integrals the function had to be continuous (which is simply wrong)). Could someone else have a look at what happens when integrand is infinite, I don't know much about that case? What do you think of "The improper integral often occur when the range of the function to be integrated is infinite."? I think it should be left out as IMO it's used more for unbounded intervals. JKBlock 14:03, 18 June 2007 (UTC)


 * I added a picture I created that depicts an integral with an infinite integrand (the integral is not convergent). I think it adds to the article to have both types depicted, but you can remove it or ask me to change the picture in some reasonable way, if you desire. Just happy to be on board! –King Bee (&tau; • &gamma;) 04:49, 20 June 2007 (UTC)

Darboux integral
Don't you think it would be a good idea to link to the Darboux integral at the definition of the Riemann integral? They are equivalent and the Darboux definition besides from being more 'strict' is relatively easy to use in proving some of the base properties of the integral (just don't try to integrate most functions directly with the definition :). I'm sorry if I "talk" to much but I need a little experience in when to edit. JKBlock 20:19, 17 June 2007 (UTC)

Removed a Harvard reference to Rudin's book
I removed a reference to Rudin's book in the beginning of the formal definition of Lebesgue integral. It was backing up a statement that the theory of Lebesgue integration is grounded on measure theory. However, this statement is hardly contested, and in having regard to the general level of foot notes / Harvard references in the article does not need a backup. Instead, the typography of Harvard references made the sentence look (to a hypothetical readed not knowing the history of the subject) like Lebesgue integral was only made possible by measure theory developed by Rudin in his book. Hence removed the reference. Stca74 20:39, 19 June 2007 (UTC)


 * Thanks for the careful read. My intent was not, however, what you infer; I simply chose an apparently awkward place to insert some kind of reference for the whole section. (Indeed, the relevance of measure theory is uncontested!) So I have restored the reference, but hopefully in a happier home. Feel free to push it around to where you think it best serves the intended purpose. --KSmrqT 00:32, 20 June 2007 (UTC)

Two remarks
I collaborate by (gently) criticising two things, hope that's OK:


 * Firstly, I don't understand the phrase "A general k-form is then a vector space with basic k-forms as the basis vectors," (in the section on integration of differential forms). Rather I'd say a k-form is an element of the vector space spanned by basic k-forms and functions as scalars?
 * Secondly, the visualization of the several approximation methods is nice, but especially the Romberg method is impossible to get an idea of by looking at the image. It's just too small. (The main article on the method gives the formulae, but not the link to the image). I would suggest to make four separate images out of the existing one, a little bit bigger and the text should refer to it (which is the case in the other 3 methods, but not that much in the Romberg one) Jakob.scholbach 15:39, 24 June 2007 (UTC)

Another thing: at the moment the advantage of the Lebesgue integral over the Riemann integral does not become clear enough. I miss a statement like: "every Lebesgue-integrable function is Riemann-integrable, and then the two integrals agree. However, there are Lebesgue integrable functions (e.g. characteristic function of the rationals) which are not Riemann integrable. Also, I assume that there are some less esoteric advantages of the extra-generality given by the choice of a measure(?). If so, this might be good to add. Jakob.scholbach 16:44, 24 June 2007 (UTC)

A reference for the fact that xx (or similar examples) has no elementary antiderivative would be good -- a reference for this fact seems to me as least as important (on an English WP article) as giving a reference for the arab integral sign. Jakob.scholbach 17:42, 24 June 2007 (UTC)


 * About your Riemann/Lebesgue comment: It should be noted that all properly Riemann integrable functions are Lebesgue integrable, but there do exist improperly Riemann integrable functions that are not Lebesgue integrable. For this reason, great care should be taken when saying things like "all Riemann integrable functions are Lebesgue integrable," just so that no one gets confused. –King Bee (&tau; • &gamma;) 18:33, 24 June 2007 (UTC)

(editconflict)


 * XX falls under Scientific_citation_guidelines. I am not sure what you mean by less esoteric, Lebesgue integration only comes up in 300 and 400 level(and beyond) math courses in college, so it might be a touch difficult to bring up a simple example as you suggest, although I believe Chan-Ho had an idea in this regard.
 * Ksmrq, will probably address your concern about the graphs. I don't quite understand your "and functions as scalars" addition, it is the 0-forms that act as the field of scalars, but I think this ought to be addressed by another editor. Cheers and thanks for your comments--Cronholm144 18:38, 24 June 2007 (UTC)


 * @King Bee: OK, I didn't know this. Then, of course the statement has to be made more carefully. But still, it ought to be there in some form.
 * @Cronholm: Well, in a way lots of things which are in the article seem to be uncontroversial knowledge. I don't disbelieve the mentioned statement. Wishing a reference was more in order to have a "further reading" where one could learn more about the fact/generalizations/explanations etc. The other thing: I wanted to say something like: a k-form is an element of the K-vector space generated by basic k-forms, where K is the field of smooth (or C^\infty) functions. Finally, by less esoteric I meant an "honest" function (not something like char. function of the rationals) possibly in conjunction with some non-standard measure, bref, something with an application outside maths, perhaps in physics. Jakob.scholbach 19:03, 24 June 2007 (UTC)
 * A practical application of the sort you seek may perhaps be found in probability theory, where Riemann integration often does not suffice. Loom91 19:50, 24 June 2007 (UTC)

Hmm... I don't know if we have have an article about functions that do not have elementary antiderivatives, we probably should, and this way the citation worry would go away. As for the honest function, not that I am aware of. Someone probably is aware of one though, and they will likely comment here soon. --Cronholm144 19:34, 24 June 2007 (UTC)


 * Jakob, thanks very much for taking the time to read and criticize the article!
 * With regard to k-forms, you are quite right that an individual form is not a vector space! Is that the essence of your complaint?
 * With regard to Romberg integration and the image, bear in mind that we use thumbnails like this in kindness to our readers with low bandwidth and/or small screens. It links to a much larger image that should have adequate detail. The text devotes a paragraph to each of the four methods illustrated. However, I confess I found it difficult to illustrate the interpolation used by Romberg in the same style as the other three methods; the choice I made really doesn't adequately convey the role of extrapolation. Perhaps I can try again after I'm finished with some other images. The vital idea of the section, both text and illustrations, is the importance of using a numerically sophisticated method like Gaussian quadrature rather than the rectangle method found in a basic caculus text. Did that come through?
 * I share your concerns about the content of the Legesgue discussion; it should clearly state why we seek a more sophisticated integral and what it buys us. I haven't had a chance to do anything about it yet, and since others have been looking at it I've been hoping someone else would beat me to it! Volunteers?
 * With regard to xx, the article could use more helpful references (and links?) overall. As well, you'd be surprised how tough it can be to come up with good examples sometimes. For instance, I'd like to illustrate improper integrals of both types (infinite range, infinite domain) with one function, and my best effort so far is x−log(x+2)/2, integrated from zero to infinity.
 * Note that an improper Riemann integral with an infinite domain can handle some oscillatory functions. For technical reasons, the Lebesgue integral does not exist if the absolute value does not converge, which is more restrictive. The Henstock-Kurzweil integral can handle any real function that either of these two can swallow, but it does not generalize to other domains as nicely as Lebesgue measure. So why is this not in the article? Maybe soon it will be; these things take experts and effort!
 * Again, thanks for your attention. --KSmrqT 22:02, 24 June 2007 (UTC)


 * 1. Yes.
 * 2. OK, the thumbnail is one thing. If I would create an illustration, I would strive for one making a clear "statement" which comes through even without a magnifier. Here, I guess, it is not even really the size which is a problem, more the 3D-ish layout of the Romberg figure, which, in this size is absolutely impossible to see, and even in the magnified view it does not become that clear, I'm afraid. Why not separate the 3 layers into 3 images horizontally side by side. I could also imagine a animated gif, if it is possible that the animation starts only when the user clicks on the image. Yes, the difference between the methods comes out, but it remains a bit obscure why the Gaussian method is actually better. "a bit of luck" is not fully satisfactory to me, but perhaps this should then be covered at Gaussian quadrature. Another suggestion: remove the list of function values and replace it by a bigger graph, where the axes are labelled. Right now, the list contributes very little to the text, only the fact that the integrals are taken from -2.25 to 1.75 is used in the text. Also, names of variables which do not occur later, like the h in rectangle method, and the T(h_k) in Romberg, should be omitted, as they are rather confusing. Finally, I think one should not take an example where certain methods work very well (or even exactly) by chance (or luck). A more detailed discussion on the subpages would probably be the good place for a double-example like the one here (first the function in the "good" range, then the "bad" range). Here, one "generic" function should be enough to show the general advantages and disadvantages of the several methods.
 * 4: Yes, WP is good in bringing all of us back to down-to-earth mathematics! e^(-abs(x)) is a little bit more compact than your function, gives improper integral 2 (-infty til infty).
 * Jakob.scholbach 01:34, 25 June 2007 (UTC)
 * Ahh, but $$e^{-|x|}$$ only gets as big as the value 1, and is bound below by 0; I believe KSmrq above was itchin' for an integral that is improper in both senses (in domain and in range). =) –King Bee (&tau; • &gamma;) 02:09, 25 June 2007 (UTC)
 * Correct. It is trivial to do one or the other; x−p is the standard example, for a fixed p. If p &gt; 1, the integral from 1 to ∞ converges but the integral from 0 to 1 does not; if p &lt; 1, the integral from 0 to 1 converges but the integral from 1 to ∞ does not.
 * Jakob, I'll try to revisit the Numerical quadrature section when I'm done with the rest. I share some of your concerns, but I had already spent a great deal of time on that one section and felt it was more important to move on to others.
 * The "accident" example is no accident; it is there to demonstrate the practical importance of insight into specific tasks. Another such example would be a periodic function, which is quite common in applications; but I felt that would be too complicated to present. In fact, I spent a long time looking for (a piece of) a function with a simple formula, a simple integral, a picture with certain properties I wanted, and so on. My intent is to use this as a running example throughout the article (except for improper integration). Consider that the current lead picture does not include negative regions, nor any use of color.
 * Thanks again for more thoughtful comments; I'll try to do them justice, starting with fixing the "forms" problem. --KSmrqT 02:44, 25 June 2007 (UTC)


 * Some additional comments: I found a nice example of a function Lebesgue can't handle. As I mentioned before, Lebesgue has problems with oscillations. Let
 * $$ F(x) = \begin{cases} x^2 \cos \tfrac{\pi}{x^2}, & 0 < x \le 1 \\

0, & x = 0 \end{cases} $$
 * Then the derivative of F exists and is finite in [0,1], but is not Lebesgue-integrable in [0,1]. For, let
 * $$\begin{align}

a_n &{}= \sqrt{\frac{2}{4n+1}} \\ b_n &{}= \sqrt{\frac{1}{2n}} \end{align}$$
 * so that
 * $$ \int_{a_n}^{b_n} F'(x) \, dx = \frac{1}{2n} . \,\!$$
 * The intervals [an,bn] are pairwise disjoint; thus if E is their union, then
 * $$ \int_E F'(x) \, dx \ge \sum_{n=1}^{\infty} \frac{1}{2n} = \infty . $$
 * This example is given in Behnke, Bachmann, Fladt, &amp; Süss (eds.), Fundamentals of Mathematics, Volume III: Analysis (ISBN 978-0-262-52095-9), pp. 462–463.
 * To challenge Riemann integration, calculate the balance point (horizontal center of mass) of a steel beam with a ball bearing resting somewhere on top. The problem, of course, is the need for a Dirac delta function to accomodate the point contact of the ball.
 * I also forgot to mention that the table used in Numerical quadrature is carefully formatted to illustrate the way power-of-two partitions with the trapezoid rule in Romberg integration can recycle evaluations, something Gauss points cannot do (though Kronrod points help). And may I just add, getting that formatting to work was a huge pain! --KSmrqT 22:02, 25 June 2007 (UTC)
 * Thanks for taking all this pain... It seems to be an ubiquitous experience, Pein in German, peine in French.Jakob.scholbach 23:25, 25 June 2007 (UTC)
 * For the sake of completeness, I suppose you should mention that while the Lebesgue integral of $$F'$$ (the function discussed above) does not exist, the Riemann integral does; it is equal to -1. –King Bee (&tau; • &gamma;) 12:48, 26 June 2007 (UTC)

In response to Jakob.scholbach's critique of the Romberg integration piece of the numerical quadrature figure, I have incorporated an exotic new element. It makes the image even busier than before, but it explicitly depicts use of the Richardson extrapolation polynomial. I'm very proud of myself for my graphical ingenuity, and for being able to explain the image in terms SVG can handle. My exuberance is tempered by a concern that the image may speak clearly to the already-enlightened, but only confuse those on the path to enlightenment. The typical numerical analysis text presents only formulas and tables of numbers, in part (I suspect) because that is the "coin of the realm", and in part because such images don't come easy.

Keep in mind that Romberg integration combines several layers of complexity. At the bottom is use of the (composite) trapezoid rule, and a formula for its error. Layered on that is exploitation of power-of-two partition refinement, which has two advantages: (1) old function values can be reused, and (2) interpolation is simplified. The third layer is Richardson extrapolation, a "deferred approach to the limit", which interpolates a Lagrange polynomial through the (h,T(h)) pairs and extrapolates to h = 0.

When I look at the image, I see all three components. Do others?

In any event, the end of the month has come, and I have not spent nearly enough time organizing and improving the many sections of text. I shall continue a little longer in that effort, and perhaps insert one or two additional images I've been working on. --KSmrqT 09:36, 2 July 2007 (UTC)

Are you working on a GIF file? The animation was rather good for pedagogy.--Cronholm144 12:46, 2 July 2007 (UTC)


 * I take it you're referring to a different image, the Riemann sum, yes? Already I have replaced the GIF with one static image, which more clearly depicts a tagged partition and the sum it generates. However, I would like to have as well an image showing increasingly finer meshes. For me, GIF animation is not an ideal solution, for several reasons: 1) no start/stop/rewind control, 2) limited color depth, 3) no resolution scaling, and in this case 4) no simultaneous view. The GIF I displaced has problems beyond that: a) it is gray, b) yet the gray areas are not solid but dithered, c) the microscopic rapidly changing number reads poorly, d) it makes comparison of fineness impossible, and e) it does not use the running example integral. Many calculus books, such as Stewart's (ISBN 978-0-534-39339-7, see also website), have illustrations that do a better job without animation. So my concept is a static image showing a series of finer divisions and the approach to the limiting value. (As SVG implementations begin to support animation better, I may begin to post SVG efforts.) --KSmrqT 22:06, 2 July 2007 (UTC)

A convergent improper integral which is improper in "both senses"
Since KSmrq above was looking for such an integral, I have one for him (and anyone else ambitious enough to evaluate it):
 * $$\int_0^\infty \frac{x^{-a}}{x+1}\ dx,\ 0 < a < 1$$

Perhaps you remember this beast from studying complex analysis. (You must integrate along a branch cut in order to work it out). Is this what you were in the mood for, or would you rather have something that can be evaluated using more elementary techniques? –King Bee (&tau; • &gamma;) 19:16, 25 June 2007 (UTC)


 * Thanks! This will do fine. A plot of 1&frasl;(x+1)&radic;x looks good, and the integral from 0 to &infin; is exactly &pi;. --KSmrqT 21:31, 25 June 2007 (UTC)


 * Glad to help out. –King Bee (&tau; • &gamma;) 21:49, 25 June 2007 (UTC)


 * Quick question - were you planning to make a graphic of the integral in question and insert it into the page? If so, where? If you don't feel like making a graphic, I can make one (similar to the other one I added to the page recently). –King Bee (&tau; • &gamma;) 12:44, 26 June 2007 (UTC)
 * Quick answer: Done. --KSmrqT 22:10, 26 June 2007 (UTC)


 * Having updated the figure for the improper integral section, I then made the mistake (?) of considering the text. I've tried to say a bit more without saying too much. One motivation is to explain the figure; another is to have enough to work with when comparing, say, Riemann and Lebesgue. I came perilously close to Cauchy principal values (within an epsilon radius?), but chose to say nothing at this point. Where are all the good editors when you need them? ;-) --KSmrqT 07:17, 27 June 2007 (UTC)


 * To be perfectly honest, I have no idea how you are computing that integral. Where is the \arctan(x) coming from exactly? –King Bee (&tau; • &gamma;) 11:14, 27 June 2007 (UTC)


 * The derivative of arctan(u) where u is equal to x-1/2. He skipped a bunch of steps.--Cronholm144 11:39, 27 June 2007 (UTC)


 * Ahh. Well, I thought that the only way to do these kinds of integrals was by contour integration, but I guess this one works out rather nicely just using the second fundamental theorem; this is why I was confused. –King Bee (&tau; • &gamma;) 11:43, 27 June 2007 (UTC)


 * You mentioned the general case of an exponent 0 &lt; a &lt; 1, which is usually nastier; but the a = 1&frasl;2 case is special and pretty. It is also sufficient for our purposes, and no one needs to see how we found an antiderivative; yet anyone who wishes can check that the derivative of 2 arctan(&radic;x) is indeed 1/((x+1)&radic;x). With this in hand, we can write closed form expressions to take to the limits, which is a Good Thing for our readers. Of course, I'm hoping that the explicit examples add to, rather than detract from, their understanding. --KSmrqT 18:16, 27 June 2007 (UTC)


 * Oh yeah, I'm all about your example. It was just early in the morning and I glanced at the Integral page and found myself completely dumbfounded by your example, since I had just worked out that integral for general a between 0 and 1 the day before. I thought you had done something amazing and beautiful, and that I could forgo contour integration; however, your choice of a = 1/2 is indeed sufficient and pretty enough for our purposes. All's well on my end. =) –King Bee (&tau; • &gamma;) 18:52, 27 June 2007 (UTC)


 * I thought I had done something amazing and beautiful — not solving the general integral, but massaging the section. ;-) For the record, Mathematica gives a closed-form antiderivative with a free in (0,1) as
 * $$ -\frac{x^{1-a} \, _2F_1(1-a,1;2-a;-x)}{a-1}, $$
 * where 2F1 denotes a hypergeometric function. Imagine throwing that at the kiddies!
 * Actually, just for cultural orientation maybe the article should give an example somewhere of how a modest little integrand can produce a scary sprawling result. Calculus book examples are seriously misleading in lulling students into thinking most integrals have tidy little closed form results; tables of integrals are not much better. --KSmrqT 22:29, 27 June 2007 (UTC)

History of integration
Take a look at Henri Lebesgue. It is not much good as biographical information, and not a great history either, but it prompted me to wonder whether we need a History of integration article. Neither Integral nor History of calculus contain any substantial information post 1850 (say), whereas a lot happened since then! Any thoughts? Geometry guy 18:04, 27 June 2007 (UTC)


 * The history section of this article might be augmented and polished. An article devoted to the history of integration might be interesting reading; it will surely be time-consuming to write. Let's do what we can here, and if it gets too long spill over. --KSmrqT 08:20, 28 June 2007 (UTC)

Example of nasty integral
In the interest of giving first-year calculus students a healthy jolt of reality, I'm looking for a really simply integrand with a really complicated antiderivative. A mild example:
 * $$ \int \frac{dx}{x^5+3 x^3+8 x^2+24} =

\frac{-14 \sqrt{3} \tan^{-1}\left(\frac{x-1}{\sqrt{3}}\right) +64 \sqrt{3} \tan^{-1}\left(\frac{x}{\sqrt{3}}\right) +26 \log (x+2)+36 \log \left(x^2+3\right)-49 \log \left(x^2-2 x+4\right)} {2184} $$ I'd like to stay away from special functions in the result, but still I'm sure we can find nastier examples. Suggestions? --KSmrqT 08:28, 28 June 2007 (UTC)


 * Thank you for this integral. I will give it to my students tomorrow. =) I will also post other nasty ones I can find. –King Bee (&tau; • &gamma;) 12:04, 28 June 2007 (UTC)

How about simply
 * $$ \int \frac{dx}{x^3+1}? $$

The contrast between the complexity of the function and of the integral is greater. Arcfrk 15:53, 28 June 2007 (UTC)

Notation in improper integral limits
As it happens, I had experimented with various ways of indicating path of approach to zero in the different limits that are used, including "+" signs, diagonal arrows, vertical arrows, and nothing. In the end, I concluded that the least clutter was the most clear, which is what I used: just plain arrows ("→") and nothing more. This is the natural expectation, and I was careful never to approach from below so the reader never has to think to get it right. The fact is, many of our readers will have no prior exposure to the plus notation; it will not help them, but only get in their way. I'm convinced it's best omitted, and have acted accordingly.

Aside from that, I delighted to see that someone besides me is actually collaborating in editing on this "COTM", especially when they catch my silly copy errors. :-)

Only a few days are left in this month, and the list of suggested improvements has hardly been whittled down at all. I have a few more contributions in the works, and certainly the article is already better than it began. But I still find it an embarrassment for what should be a project showpiece. --KSmrqT 11:25, 28 June 2007 (UTC)


 * This sets a bad precedent. The limit notation with pluses and minuses is standard, and we should report on it rather than skirt the issue. Further, you said that you were careful never to approach from below so the reader never has to think to get it right, but again, this is not a textbook where we are free to try our innovative pedagogic approaches. We should be up front about the issues, just avoid making them unnecessarily complicated. Arcfrk 16:02, 28 June 2007 (UTC)


 * I understand your dismay, but the suggestions at the top of the page are horrendously vague. "The computing integrals section could do with some expansion"? What does that even mean? We have main articles on all the techniques that one would learn in a calculus course. Should we have more examples? Use other, more advanced methods than the second fundamental theorem? Or should I just be bold and start going nuts? –King Bee (&tau; • &gamma;) 12:03, 28 June 2007 (UTC)

If you truly go nuts do it in the sandbox, see what happens--Cronholm144 12:20, 28 June 2007 (UTC)

A-class review
An A-class review (a good way to rope in more editors) seems appropriate now that the end of the month is coming up, thoughts?--Cronholm144 21:52, 28 June 2007 (UTC)


 * I think an A-class review would be premature. The article is struggling to be B-class, much less Bplus. For example, the content, organization, illustrations, and references (not yet one per section) all need work. Ask yourself, is it:
 * ► correct? ► reasonably complete and balanced? ► clear? ► compelling? ► grammatical, correctly spelled, and well typeset? ► appropriately illustrated, where applicable? ► well linked? ► helpful in providing references and additional resources? ► reasonably accessible, given the topic?
 * Much has been accomplished; much remains. Imagine that your precocious young daughter becomes curious about integrals, and comes to Wikipedia for her first exposure. Is this the article you want her to see? Will she be oriented, educated, entranced, inspired? OK, so that's a bit much to strive for; but I know the article does not yet satisfy my A-class goals.
 * I still have a little time before the end of the month; I shall try to invest it well. --KSmrqT 07:14, 29 June 2007 (UTC)

Well, the end of the month isn't going to be the end, it will be A class one way or another. :) --Cronholm144 11:41, 29 June 2007 (UTC)


 * A precocious young daughter of mine getting interested in integrals? This sounds like a dream I once had; it was a good dream. =)
 * I will continue to think of ways to improve the article. –King Bee (&tau; • &gamma;) 13:15, 29 June 2007 (UTC)

The idea of me having a child of any age is frightening to me...--Cronholm144 13:47, 29 June 2007 (UTC)

Sorry I haven't been more help of late (I had a burst early in the month), but I've been busy moving house. I do feel the article has been significantly improved, if nothing else then in terms of structure and fuller content for important sections. It does still need a lot of tweaking, and some conscientious referencing certainly wouldn't go astray; still, moving from start to class to B or B+ is not bad for month's effort. -- Leland McInnes 15:47, 29 June 2007 (UTC)

The lead
KSmrq, I'm afraid your edits are simply incorrect.


 * 1) Refinement is not the correct word to use. Integration extends the concept of summation. Refinement refers to taking a subset, which is obviously not the case here.
 * 2) Functions don't need to be continuous to be integrable, and that's that. Continuity is not a rquired condition in any definition of the integral, even the Riemann one. It's perfectly possible to integrate step functions. I suggest you take a look at Apostle's classic text Calculus.
 * 3) The paragraph you removed was the only one that said anything about what an integral was. Unless you have a replacement, you should not remove it. Your lead does not tell the reader anything about what integration does in general.
 * 4) "and dx denotes the weight in the "weighted sum", multiplying a height, say, into an area. (Introductory courses may treat dx as merely denoting the variable of integration.)" I don't know where you got this, but I can't find any mathematical meaning in it. Wikipedia is not the place for original research, please don't put in such strange statements unless you can source them. For a Riemann integral, dx does denote the variable of integration. It does so not only in introductory courses, but for graduate courses, doctoral students and professional mathematicians also. There's nothing anyone can do about it. In Lebesgue integral, it denotes a measure. Your statement is meaningless. Again, I suggest you take a look at Apostle.
 * 5) Why do you insist on giving an incomplete statement of the fundamental theorem in the lead? Such auxiliary material does not belong in the introduction, they should go to the main body of the article. Loom91 07:09, 5 July 2007 (UTC)


 * I'm sorry your grasp of the English language and of integral calculus and of helpful writing is so limited, but please do not punish Wikipedia. You have repeatedly insisted on "correcting" things that are not mistakes. I'm afraid I will insist on reverting, and probably wasting my time explaining (as others have) precisely why.
 * The American Heritage Dictionary offers multiple definitions of "refine", none of which agree with you. One is "to acquire polish or elegance." Refinement is, of course, the act of refining; but also "a keen or precise phrasing; a subtle distinction." Ironic that you don't understand that. And were we to be using the word in a formal mathematical sense (which clearly is not appropriate here), you are still wrong; refining a partition for a Darboux sum is not removing a subset, to give one example.
 * This is now the nth time you have pulled this bullshit about continuity. Each time, the editor in question has not been saying what you assert in your reversion. Nobody has ever said we can only integrate continuous functions, nor only functions over continuous domains. We try to begin with something basic, and if you would look at a variety of calculus texts you will find that every author does so, as is only good pedagogy.
 * The removed paragraph said almost nothing; and what it did add, I incorporated as one sentence in the preceeding paragraph.
 * Again, it is you who needs to learn. Every definition of integration requires a combination of the function value and something more or less equivalent to measure, which is explicitly a "weight" in a "weighted sum". For the Riemann integral computing area under a curve, it comes from the width of the intervals in the Riemann sum. For the Lebesgue integral, say of a simple function, the measure is used precisely as a weight in a weighted sum. You seem unable to see what is right in front of you.
 * There are many variations of the fundamental theorem, and each has technical conditions. In fact, one of the motivations for defining different integrals is so that it can hold in greater generality. The lead is hardly the place to go into that. But mention of the fundamental theorem absolutely must go in the lead, as it is one of the most important facts about integrals that anyone should know.
 * That's a point-by-point rebuttal. And your insults are, well, a poor reflection on you. Enough. I'm reverting. (But I'll keep and correct your "Apostle" citation, which I had planned to add along with several others myself.) --KSmrqT 12:36, 5 July 2007 (UTC)


 * I would like to mention tat the removed paragraph mentioned in point 3 does, indeed, say something, and would like to lobby for its re-inclusion, even if it is in a slightly cut down form. What I am looking for here is some material in the lede that provides a very accessible description of integration -- if you like a summary of the "Informal discussion" section. Currently the lede is quite concise, but fails to provide much for the truly mathematically naive reader -- something I think we can and should do, especially if it will only cost us a couple of sentences. -- Leland McInnes 13:47, 5 July 2007 (UTC)


 * I realized that's what the paragraph was trying to do, but it didn't succeed. For reference, here's what I removed:
 * 
 * Integration is used to find the "total amount" of a quantity that varies across a given domain. For example, the velocity of an accelerated body  changes instant to instant through the continuous domain of time. To sum up all the instantaneous velocities over a given interval of time, and hence obtain the total displacement that occurred, we evaluate the integral of the velocity over the given interval of time. Though this concept was the starting point for the development of integration theory by Newton and Leibniz, it has since been extended, and newer definitions stress different aspects.
 * 
 * Despite repeated tinkering by different editors (including me), it ended up using a lot of words to little benefit. Frankly, I feel the same about the sprawling "Informal discussion". We absolutely do need friendly, gentle material early, but I'm not satisfied with what we've got. If someone else would be so kind as to fend off Loom91's misguided revisions, I can concentrate on my project of doing something about this, maybe with illustrations. If you like, I can go into more detail about my complaints, but I think my time is better spent rewriting so you can see what I'm trying for. --KSmrqT 14:52, 5 July 2007 (UTC)


 * I agree that the tinkered with paragraph (it started out a little shorter I think) is not really quite what we want. I feel we should be improving it rather than removing it however. I would be more than happy to work with you on this -- one of my main interests in this article is ensuring it is accessible to a general audience. In that sense I would be happy with some discussion rather than just tinkering/changes. One of my major frustrations when first working on the lede was that people would simply tinker particular points that they wanted changed with little consideration for a coherent whole. I feel we woul do well to come to some agreement as to generally what we would like to say, along with ideas as to how best to say it, and draft it as a whole, rather than piecemeal tinkering at this stage.


 * With regard to the informal discussion: that is my work, though I still consider it somewhat of a draft. I spent some time working out generally what I wanted to say, and then sat down one afternoon and tried to write it as clearly as possible. The result, I admit, is a little sprawling, but I feel that the informal discussion is in many ways at least as important as any other, and deserves some space to decently communicate the underlying ideas. I would be happy to hear and discuss your criticisms of it (I have some myself that I haven't gotten to fixing yet) and hope that, with some discussion we can move toward something better. -- Leland McInnes 15:46, 5 July 2007 (UTC)


 * Sorry, can i just interject to ask where the grammar has gone on this article? A lot of the prose is very choppy, and doesn't flow AT ALL throughout the article. Also, it's become just like other articles where chunks of information are thrown onto the article and left. Please, tell me if there's any concensus to keep this way, otherwise i'd very much like to clean it up. ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ slurp me! 15:31, 5 July 2007 (UTC)


 * I certainly would like to clean the whole article up for flow, but was waiting for it to stabilise on overall content a little first. Feel free to get started however. I certainly agree it could use some work. -- Leland McInnes 15:47, 5 July 2007 (UTC)


 * It would be a mistake to work on wording for flow just yet, when major sections may appear and move around. See below, for example, where we're discussing injecting a substantial section on complex integration. I'm going to be working over the section on Lebesgue integration, and some of the others are really more like rough drafts. If you look at the rating (top of this talk page), this is still just at start class, not even B, so it's considered a diamond in the rough, emphasis on rough. Thus if you come back in a week, big pieces of the article may look rather different. In my view, we're still at the grinding stage, and it would be a waste of time to start fine polishing. That said, I think there is more good content than two months ago.
 * What would be helpful now is to skim and report overall, and section by section: What do you notice that is missing, confusing, out of order, or possibly superfluous? Which sections are in decent shape, and which not?
 * For example, the history section has no business interjecting itself between formal definitions and properties, but there it sits — for the moment. The Lebesgue integral can't handle an improper Riemann integral discussed earlier on this page, which should be brought up in the Lebesgue section, except that improper integrals are currently much later in the article. To give two examples.
 * Did you just come late to the party? This was the Collaboration of the Month for June, and now you are kvetching? Ah well, better late than never, I suppose. And, honestly, much of the article hasn't been touched yet. --KSmrqT 16:51, 5 July 2007 (UTC)
 * I will be honest here -- at the moment, it's not too pretty. A lot of the grammar even for experienced Mathematics majors would seem somewhat vague and nonspecific. Even all those years ago, before I went to university my tutors told me that the way that you should portray information if you want to be heard is to speak with clarity understood by everymen.
 * I still think it's a good idea to have a sufficient curve into the material to allow people who are unfamiliar with the subject to understand it, rather than for us to focus on the semantics of whether it's correct or not -- focus on the accessibility of the actual article for the majority of the accuracy, and we should just agree on a middle-ground for terminology.
 * I have free time tomorrow, so I'll work on it then. For now, let's just be careful with the article. ♥♥ ΜÏΠЄSΓRΘΠ€ ♥♥ slurp me! 20:31, 5 July 2007 (UTC)

The first and the last points are not important, let's talk about the other three.
 * For a Riemann integral, dx denotes the variable of integration. I've provided a reference to the exact line of a textbook where this is explicitly said. I don't believe you have any source for the strange statement that it is a weight in a weighted sum.
 * I assume by the most basic integral you mean the Riemann integral. A function does not need to be continuous to be Riemann integrable. Step functions are Riemann integrable (reference Apostle). If you are using some different definition of the most basic integral, please point that out. I've looked at several calculus textbooks, and none of them assert that a function needs to be continuous to be Riemann integrable.
 * I don't know what sentence you are referring to. That paragraph may not have done a good job of giving an intuitive idea of what an integral is, but currently the lead does no job at all. An effort, even a poor one, is better than none. Write a better introducion, but untill you do, please don't remove the one that was existing. Loom91 12:56, 6 July 2007 (UTC)
 * Please stop this.
 * Calculus books are written at many different levels for many different kinds of students. Collectively, they say all kinds of things, including falsehoods. Do you not know what a weighted sum is? Do you really need a textbook citation to believe that a Riemann sum (for example) is a weighted sum?!!
 * If you understood integration more deeply, you would realize that the Riemann integral is critically dependent on a certain amount of continuity in the function itself; thus the Dirichlet function is not Riemann integrable. The first paragraph is not the place to go into technical details about isolated points of discontinuity; and the typical elementary examples are invariably continuous functions with continuous derivatives. You come up with a way to twist the interpretation of common language to dispute an unremarkable statement. Give it up.
 * A quick click on the "Compare" button would answer your question. But since that seems to elude you, the sentence I added to the first paragraph (which also addresses your "continuity" nonsense) is "More sophisticated integrals can handle functions and domains of even greater generality."
 * You are simply making a nuisance of yourself, and not contributing productively to the development of the article. You have a habit of doing this, and have been warned about the same misbehavior at other articles. The fact that you misspell the well-known Caltech author "Apostol" as "Apostle" suggests that you are pretending to more expertise than you really possess, both in calculus and in English. So does the fact that you don't even bother to adhere to the article's Harvard style when you try to insert a reference. And the fact that you have written nothing to fill in missing sections of the article suggests you're really not interested in contributing. Therefore I suggest you go find something more productive to do with your time. (And by the way, I have written a better introduction, as well writing broad swathes of other content and producing numerous illustrations.) --KSmrqT 13:58, 6 July 2007 (UTC)

Now you are violating several Wikipedia policies. You are engaging in revert war without participating in talk page discussion, you are being uncivil by calling edits you disagree with "bullshit", removing explicitly sourced content without providing rationale, and inserting unreferenced original research. All of these are against core Wikipedia policies. Please engage in civil discussion and provide references for edits. Loom91 13:30, 6 July 2007 (UTC)
 * You are mistaken. I took the time (just above you, if you will only look) to compose a thoughtful reply. You couldn't wait to read it. --KSmrqT 14:02, 6 July 2007 (UTC)


 * In top of being uncivil, you are now lying. You wrote that reply long after you reverted me and inserted it between my two comments to make it appear as if I had not read your comment. This is not a good way of becoming a good editor. I'm afraid that the Wikipedia policy of providing references is not negotiable. If I provided a reference saying the Earth is square and you couldn't provide a source saying the contrary, my edit will stand. The fact is that I've provided a reference and you haven't. Stop reverting or I'm going to start a RfC. I assure you that other editors place even more emphasis than me on citing sources. Following policy, particularly core ones such as providing references and not doing original research, is not optional under any circumstances. What you or I do or do not believe has nothing to do with what stays in the article (and neither does expertise or qualifications, yours or mine). If you fail to agree with it, I'm afraid Wikipedia is not the correct place for you. May I suggest you carefully go through the Pillars of Wikipedia? A piecewise continuous function is not a continuous function. Making it so is not a simplification, it' an error. Your undoubtedly valuable contributions do not empower you to escape the policy of citation. Once and for all, provide references or cease this revert war. Loom91 16:17, 6 July 2007 (UTC)


 * I wrote my reply immediately, though it took some time. I took no steps to "insert it", but simply posted it in the usual fashion, immediately after your last remarks visible to me at the time.
 * I'm not impressed by your WikiLawyering. Describing a Riemann sum as a weighted sum is a trivial observation, and if you continue to attack it as demanding a citation you will only make yourself look (more) foolish. (For example, I note your arrogance towards Petergans, where you denigrate a university chemist with considerable expertise, and the way you attacked JRSpriggs' writing at Newton's laws.) And you persist in mischaracterizing the statement about continuity, in ever new ways, but always with your predetermined conclusion. You acknowledge my contributions, yet presume to lecture me. Your threats about citations are especially ironic in light of the fact that almost every citation in this article was put there by me, often with considerable digging to find the sources.
 * Meanwhile, you have yet to make any positive contributions to this article. Based on behavior, you have no interest in building an encyclopedia, only in making a nuisance of yourself. Well, congratulations; you're doing a fine job. --KSmrqT 17:33, 6 July 2007 (UTC)

In an effort to bring some reasonable discourse back to this discussion... I've been thinking about how to provide some sort of intuitive/accessible description of integration for the introduction (really we only need a sentence, but something more would be very helpful). I wish to avoid "area under curve" descriptions because ultimately they are rather simplistic. Instead, going with the "weighted sum" description alredy present, I was hoping for something along the lines of:


 * Intuitively an integral may be thought of as a sum of an infinite number of infinitesimally weighted values.

The only sticking point with this for me is that "infinitesimally weighted" is perhaps a little too dense for an accessible description, something like ...of an infinite number of values, each given an infinitesimal weighting. One could raise objections about bringing infinitesimals into things, but they provide a more intuitive approach, albeit not a formally justifiable one (barring orking in smooth world toposes or similar). Thoughts? -- Leland McInnes 15:06, 6 July 2007 (UTC)
 * I agree that the current first paragraph is still less than ideal. We can try to improve it, but with so much major work pending for the rest of the article, I'd prefer to polish elsewhere just now. Note that article leads, especially first paragraphs, are popular battlegrounds. Why? Here's the implicit job description: "Describe Topic X in one sentence, which must be meticulously correct, totally complete, and easily understood by a great ape." :-) (Have you ever thought of what it would be like to write definitions for a dictionary? That's a real challenge!)
 * But since you ask, specificially about infinitesimals, I think that steps in a tiger trap. It's a technical detail about how integrals might be calculated. Too soon! What the reader wants to know in the first paragraph is, what is an integral, and why should I care?! From the inside (our view), your sentence is an answer to "what"; but from the outside (the lay reader's view), there is nothing intuitive about an infinite sum of infinitesimals. --KSmrqT 15:39, 6 July 2007 (UTC)


 * I understand a certain reluctance with regard to infinitesimals, but I think we need to say something, and this represents a simple explanation. We need to say something about what an integral is in the lede, and currently there is nothing adequate there "sums of continuous functions over continuous domains" is far from satisfactory and the example is a touch technical with its terseness (you and I know full well what is meant, but it may well be less clear to those with, for example, a limited grasp of functions). I don't feel that the description given is "how integrals might be calculated", but rather a description of what simple integrals are -- you yourself are calling them "weighted sums", this merely expands on this to provide a description of a continuous/smooth weighted sum: one which sums infinitely many values, each of which is given an infinitesimally small weighting (since each represents an infinitesimally small part of the total sum). -- Leland McInnes 17:41, 6 July 2007 (UTC)


 * One final remark, about continuity. Purely to establish a matter of fact, I quote the very first definition of integral from Keisler, Elementary Calculus: An Approach Using Infinitesimals, ISBN 978-0-87150-911-6, Chapter 4: Integration.
 * 
 * Let f be a continuous function on an interval I and let a &lt; b be two points in I. Let dx be a positive infinitesimal. Then the definite integral of f from a to b with respect to dx is defined to be the standard part of the infinite Riemann sum with respect to dx, in symbols
 * $$\int_a^b f(x) \, dx = \mathit{st} \left( \sum_a^b f(x) \, dx \right) . $$
 * 
 * Not happy with an ultra-modern approach like non-standard analysis? Fine; try Stewart, Calculus, 5th ed., ISBN 978-0-534-39339-7, &sect;5.1 (one of the most heavily-used standard textbooks).
 * 
 * Definition The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles:
 * $$A = \lim_{n\to\infty} R_n = \lim_{n\to\infty} [ f(x_1) \Delta x + f(x_2) \Delta x + \cdots + f(x_n) \Delta x ] $$
 * 
 * Furthermore, the next sentence emphasizes the use of continuity: "It can be proved that the limit in Definition 2 always exists, since we are assuming that f is continuous." Both authors contradict claims in this discussion that no basic definition of an integral requires continuity. And note that Stewart is an author with decades of feedback from students and teachers about what works and what doesn't in teaching, in a market with tremendous competition.
 * Is this the only approach? Of course not. Other excellent authors build the ideas and definitions in other ways; it's a big market. Some whom I respect are Tom Apostol at Caltech, Gilbert Strang at MIT, and especially Richard Courant and Fritz John. (And for analysis, stalwarts like Walter Rudin at U. Wisconsin, and Andrei Kolmogorov.)
 * The claim at the start of this thread was "Continuity is not a rquired [sic] condition in any definition of the integral". Well. If there is any more big talk about "simply incorrect", perhaps it should come from a more credible source.
 * Though there was never a serious question of correctness, the question of helpfulness remains open for discussion.
 * Since I enjoy reading the masters, I'm always curious to see how they approach things. Kolmogorov &amp; Fomin, ISBN 978-0-486-61226-3, on page 293 introduce the Lebesgue integral with these words: "The concept of the Riemann integral, familiar from calculus, applies only to functions which are either continuous or else do not have 'too many' points of discontinuity." Schmetterer and Stender, in Chapter 3 of Behnke, Bachmann, Fladt &amp; Süss (eds.), ISBN 978-0-262-52095-9, on page 60 quantify "too many", as follows: "The fact is that a function g &isin; B(J) is Riemann integrable over J if and only if the set of points of discontinuity of g in J is of Ln-measure zero." (Here J is a closed interval in Rn and B(J) is bounded functions over J.) Well, sure; everyone understands that! Not.
 * Thus, as I said much earlier in this thread, the Riemann integral does assume a certain amount of continuity, and it would be wildly inappropriate to explain more in the first paragraph. But I see now I might as well treat this like split infinitives or "which" versus "that": avoid saying things that are perfectly correct so I don't have to waste time arguing.
 * Finally, I do apologize for being perturbed enough to interject an expletive, accuracy notwithstanding. I plead fatigue: I've been working hard on this article for over a month, and ran out of patience. --KSmrqT 11:33, 9 July 2007 (UTC)

I may not be a mathematician, but I do know a little math. I never said that functions do not need to be continuous at all to be Riemann integrable, only that being everywhere continuous was not necessary. As you say, the concept of piecewise continuity can not be introduced in the lead, but neither does one need to give the impression that only continuous functions are integrable. I was advocating the solution arcfrk has come up with, simply removing the word continuous. Therefore, I don't think this debate is relevant anymore unless someone reinserts the word continuous in the lead. Loom91 21:10, 11 July 2007 (UTC)

Contour integration/residue calculus
I have a question. Since use of the second fundamental theorem of calculus and the techniques one would learn in an undergraduate course are seemingly given a lot of consideration in this article, should we more thorougly cover the use of residue calculus to evaluate improper integrals? It could fit nicely in the improper integrals section, and I would be willing to write up an example and contribute images. However, if this is beyond the scope of what this article should contain (since this is a particular, albeit standard method), then I will not add it. I just want some opinions here. –King Bee (&tau; • &gamma;) 12:59, 5 July 2007 (UTC)


 * My initial concern is that this would result in the section on improper integrals being significantly larger than any other section, which certainly seems rather unbalanced under the circumstances. Some mention of residue calculus may not go astray, but I think we need to consider how best to work it in. Perhaps a short section on integration in complex analysis? -- Leland McInnes 13:40, 5 July 2007 (UTC)


 * Ah, it seems a few of us are still around feeling the need for more improvements. Good. My reading of the article is that it says very little technically about the fundamental theorem; we have a little overview, a little history, but nothing of substance. We also say almost nothing about complex analysis. So, I would like more of both. However, I agree with Leland that the improper integrals section is already large enough. It's already twice as long as the Riemann integral section (even though the latter now has a figure for convergence where I worked in a previous request to mention the Darboux integral)!
 * Singularities are important in theory, but to treat them properly we must work with complex integration. Why? Because a singularity in the complex plane, not on the real line, can cause problems for a purely real radius of convergence. And that gets us into complex analysis, which really deserves a section. So that's where you discuss residue calculus. --KSmrqT 14:31, 5 July 2007 (UTC)


 * So, the impression I'm getting is that it would be a good idea to expand in this manner? I'm going to start whipping up some pictures tonight, and get going on some prose to accompany them. –King Bee (&tau; • &gamma;) 00:05, 6 July 2007 (UTC)


 * Yes, it is a Good Idea to introduce a section on complex integration, including a discussion of residue calculus.
 * I'm afraid that, so far, my contributions have not been paragons of economy; but the idea is a brief overview and orientation for each subtopic, leaving the bulk of the work to the specialized article.
 * Good luck with the pictures. I've seen some nice 3D depictions of poles and Riemann surfaces, but usually those take noticeably more work to produce. Before investing too much time, you might check Commons to see if there's anything we can appropriate. --KSmrqT 14:37, 6 July 2007 (UTC)
 * Thanks for the advice. As far as the pictures were concerned, I was thinking something along the lines of examples of contours one might use to evaluate improper integrals of functions of a real variable. I've also been working on a better picture for the 3-D integration (rather than just the volume of a parallelpiped) that has a paraboloid or something like that. I should be making relatively sane progress throughout the weekend. –King Bee (&tau; • &gamma;) 15:19, 6 July 2007 (UTC)
 * One obvious place to look in Commons is Category:Mathematical analysis, but images are often poorly categorized. And the web is always a rich source for examples.
 * When making images, I would recommend keeping text to a minimum. A minor (though important) reason is to reduce clutter. A major reason is to increase reusability, both in other articles and in other languages. --KSmrqT 16:00, 6 July 2007 (UTC)

State of the article
It was suggested that we should take an overall look at the shape of the article: the state of each section, what is missing, what is too long, etc. Let's get started on that. Fell free to add comments below.


 * 1) Lede
 * 2) * Better than it was, but I feel it could use an eye to accessibility. Leland McInnes 19:49, 5 July 2007 (UTC)
 * 3) *The term "weight in the 'weighted sum'" is not defined or explained before. To me, it is rather distracting. At this point, I would only give the "introductory course" explanation, referring to the differential forms and measure sections for the actual explanation. (Most readers won't need to know this in the first place). The explanation of the terms integrand, domain of integration might be made shorter by something like
 * $$\int_{\color{Blue}a}^{\color{Blue}b} {\color{Red}f(x)}\,dx . $$
 * Possibly these things might also be merged somehow with into a "notation" section (outside of history, though).Jakob.scholbach 23:48, 5 July 2007 (UTC)


 * 1) Informal discussion
 * 2) * Still a draft, but contains most of the useful ideas; could use diagrams, and judicious cutting down. Leland McInnes 19:49, 5 July 2007 (UTC)
 * 3) Formal definition
 * 4) * I'm reasonably happy with this, it could use some polish, but the core is in order. Leland McInnes 19:49, 5 July 2007 (UTC)
 * 5) *The section on other integrals should present for every type a short (one-phrase) description of the various types, otherwise it has more the character of a see-also-list. Several integrals are linked more than once. The phrase with the steel ball resting on a beam is incomprehensible (to me). A graph illustrating the intuition behind the Lebesgue-integral (something like this was discussed I remember) would be good (use a non-standard measure to illustrate Lebesgue's power!). Jakob.scholbach 00:06, 6 July 2007 (UTC)
 * 6) History
 * 7) * Is a little out of place, but postponing it to the bottom was even worse. This is an important accessible section of the article, and burying it won't help non-technical readers. It needs to be up front somewhere. Content wise it covers the important points, though the post N&L section is light. Leland McInnes 19:49, 5 July 2007 (UTC)
 * 8) Properties of the integral
 * 9) * I feel this hits the major points, but is a little terse. Concision is nice, but a little discussion wouldn't hurt. Leland McInnes 19:49, 5 July 2007 (UTC)
 * 10) *Concerning the basic inequalities: an image "proof" for the first given inequality would be nice (and easy to do and would not consume too much space). I miss the word "Schwarz inequality". The "Conventions" are IMO not a property, but rather a notation-related piece of information. Jakob.scholbach 23:48, 5 July 2007 (UTC)
 * 11) Extensions
 * 12) * Is missing a discussion of integrals in complex analysis (which was in the original sandbox plan), and the multiple integrals subsection could really use some work. The rest is not bad, though the differential forms subsection could use some trimming down.
 * 13) *I disagree on trimming down the differential forms here. Rather I'd treat it as the first extension and try to explain or relate the other ones to diff. forms. The step from a "simple" integral to one with several variables (and also line integral-surface i.) is conceptually easier than differential forms. Therefore this deserves a pretty thorough discussion even in the integration article, I think. Jakob.scholbach 23:48, 5 July 2007 (UTC)
 * 14) *Where is non-classical analysis? --KSmrqT 00:26, 6 July 2007 (UTC)
 * Ugh, yes. We'll be getting awfully long if we do much more than skimming. I can write something on smooth infinitesimal analysis (I have Bell's text on this on hand at the moment) and could write soemthing to p-adic analysis, the rest is stetching beyond my field. -- Leland McInnes 00:36, 6 July 2007 (UTC)
 * I think one sentence and the link would suffice; just say something. --KSmrqT 04:05, 6 July 2007 (UTC)
 * Perhaps its my soft spot for smooth infinitesimal analysis, but it may be worth saying a little more, since it provides such an elegant approach to integrals. -- Leland McInnes 04:52, 6 July 2007 (UTC)
 * 1) Methods and Applications
 * 2) * Should probably come before Extensions; from a pedagogical perspective it is the easier of the two sections. In general I am unhapopy with the "Computing integrals" subsection, which is comparatively quite terse and lacking in detail. Leland McInnes 19:49, 5 July 2007 (UTC)


 * 1) Missing Material
 * 2) * Integration in complex analysis, residue calculsu, etc. Should go in "Extensions" after multiple integration. Leland McInnes 19:49, 5 July 2007 (UTC)
 * 3) Excessive detail
 * 4) * Nothing as yet, though the article length is getting long, so some cuts may have to be decided on in future. Leland McInnes 19:49, 5 July 2007 (UTC)

Introduction (was "Informal discussion")
As promised (threatened?), I did not make piecemeal changes to the informal discussion; I wrote a new one. It's even longer than its predecessor, and has a new name. However, I hope it makes up for that by saying more. Especially, I hope it is more gentle and helpful for the typical reader of this article.

It calls for two figures. The easier one shows two stepwise approximations to
 * $$\int_0^1 \sqrt{x} \, dx, \,\!$$

with five steps and right-end samples, and with seventy steps and left-end samples. These happen to be Darboux sums, so can be reused for that article. The harder picture is a 3D (or pseudo-3D) depiction of the swimming pool example, to illustrate integrals of volume, surface, and curve. I haven't yet thought much about its appearance.

But before I invest that effort, I'd like some feedback to know if the new section as a whole, or at least that example, is likely to be retained. Unfortunately, the response to calculus writing seems to vary widely, depending on the reader. (Read some reviews at Amazon for a taste of what I mean!) One reader may love it; another may hate it. The question is: will everyone hate it?

One last point: Leland McInnes invested time in writing the prior incarnation, and I promise I did read it more than once, and did try to draw inspiration from it. I used none of those words and none of those examples, but the good in it lives on. If that was a first draft (as was said), then this is a second draft.

Enjoy. --KSmrqT 15:15, 6 July 2007 (UTC)


 * I'm working on your sqrt(x) picture right now. Should I keep the theme with that cool blue color you have thrust forward (for the shading of the rectangles), or should I switch it up? –King Bee (&tau; • &gamma;) 15:51, 6 July 2007 (UTC)


 * Actually, I've got it in progress, using the same machinery I used for the previous illustrations. (I would have it done already, if I'd stop talking!) My originals are SVG; unfortunately the software MediaWiki has chosen is a painfully deficient implementation, so I oftne upload PNG instead. But for reference, you can find my standard four colors at the bottom of my user page.
 * These are not arbitrary choices; they are experimental results from studies of the human visual system, so-called "unique hues". As data makes its way from the retina to the brain, it is encoded using an opponent process. The most important axis is black/white; it also has the highest resolution. Next is red/green; and last is blue/yellow, which has noticeably lower resolution. This influenced the design of the analog television upgrade from black-and-white to color. It also shows up in the Berlin and Kay investigation of basic color terms in languages around the world. So my blue is a really special blue, not just a tasteful choice. :-) --KSmrqT 16:38, 6 July 2007 (UTC)


 * Wow, and here I thought it was just a really pleasant shade of blue. =) (As a note, the other picture I contributed used the same hexcode as your particular shade.)
 * Well, if you have the picture in progress, maybe I'll just let it be, and work on other pictures. I also create images in inkscape, and they are svg; and I upload as png for the same reason you described. –King Bee (&tau; • &gamma;) 17:47, 6 July 2007 (UTC)

I have to admit to being a little unhappy with this new version; in part it is a little long, and in part I feel it fails to highlight what I feel is the important distinction of integration, which is resolving the issue of the continuous. I'll try and proviode more detailed comments on this, and suggestions for improvement, at a later date. -- Leland McInnes 18:26, 6 July 2007 (UTC)


 * I am more than a little unsatisfied with the introduction (formerly, informal discussion). The main problem seems to be the textbook tone it takes, together with the substantial increase in sophistication after the last revision. Remember: the goal of this section is not to summarize what we know about the integral! That honour belongs to the lead. Rather, it should be a gentle motivation for integration and integrals, if indeed that is possible. In particular, any types of numerical examples, and especially, worked out problems, appear extremely unencyclopaedic, to the point of violating WP:NOT. Looking over the recent attempts at the informal introduction, I've begun to think that, perhaps, including it may not be a good idea, at least, if we want this entry to read as an article in an encyclopaedia. Opinions? Arcfrk 02:00, 7 July 2007 (UTC)


 * I feel that something in the way of an informal discussion is important for WP:MTAA. Integral is an important article and should be one of the foundation articles regarding calculus, thus it should provide material for interested readers with no understanding of even basic calculus concepts, ideally as close to the beginning as possible. Technical material can be deferred till later in the article. I certainly agree that the aim should not be to summarise everything we know about integrals. The aim should be to provide some semblance of discussion of integrals for a general audience. I felt my earlier version was too long and could be usefully cut down to the short discussion required. I feel the current version is far too long and covers far too much material. Given the nature of the revised version I feel we should perhaps start with some discussion of exactly what we would wish for the section to communicate, since apparently I and KSmrq have rather different ideas as to what should be discussed. -- Leland McInnes 02:26, 7 July 2007 (UTC)

The first problem I perceieve with the current introduction (formerly "informal discussion") is that it is just too long, often taking much time to discuss topics beyond its scope. For example, the second paragraph is long and essentially purely about the issue of incommensurable lengths, and the real number line. While some discussion is useful in that addresses the reasons for the early incompatability of discrete and continuous worlds, it is well off topic about integrals.

The second point I want to discuss is the amount of semi-worked examples. This is not a textbook, and worked examples are not required; in practice they simply extend the length with the necessary explanations of particulars.

My third issue is with the relative level of technicality, and the degree of mathematical sophistication required. While the sesction begins with a very general tone, it is not long before we are blithely tossing around functions with little or no explanation. If this section is to be a description for general audiences that I feel it should be we should be following the Hawking view that "every equation halves the readership" and try keep things at a low level. When the introduction continues and gets into Lebesgue integrals and differential forms I feel we are extending well beyond what can be adequately covered in such a section.

My essential point is that it seems that I and KSmrq have rather different ideas about what this section should be. This probably needs to be fixed before we can proceed. Towards that end, let me make clear my own views on what the section should contain so that we can work toward some consensus or compromise position. I feel the role of this section is to provide a relatively concise description of what integration is for a completely general audience. That is, where we should provide a one sentence "intuitive" description of integrals in the lede, this section should be nothing more than the expansion of that sentence to a paragraph or two. In short it should be the informal definition (being very informal and high level).

What specifically do I feel the section should discuss? I don't feel anything more is required. Equally, I feel that these points should be addressed as directly as possible.
 * The opposition of the discrete and the continuous.
 * A high level view of how integration extends/refines discrete sums into the realm of the continuous.
 * Some manner of physical example showing how this is useful; be it areas, or rates of change -- the key point being that it deals with continuous worlds (geometry, or time) rather than discrete ones.

I hope this will help us move forward in discussion of how to handle the "introduction" (formerly the "informal discussion"). -- Leland McInnes 17:26, 7 July 2007 (UTC)


 * I'm quite happy with the introduction as it is now. First, I think it is not too long. The subject is not small enough to give a short intro without cutting essential parts of the story. IMO this is exactly the place where a comparing glance over the different types of integration should take place (as it is done right now). A reader with no or little knowledge of calculus needs to be able to "understand" the part up to the Darboux sums, which I think he could. A high-school will read until he meets his friend(?) "Fundamental theorem of calculus". I would add "the related function F(x), the so called antiderivative", though. Finally, a undergrad/grad student will be able to read up to Lebesgue and Stokes. So all kinds of readers find a piece of information fitted to their level/interest.
 * I agree with the previous poster that the opposition of discrete and continuous is important - it seems to be there, namely in the "In the simpler case where y = 1, the region under the 'curve'". However, a different wording may enlighten the intention more. Perhaps compare summing sqrt(0)+sqrt(1)+sqrt(2)+sqrt(3)+sqrt(4) to the integral of the function from 0 to 4. Even better would be to find some physical phenomenon (like an accelerating rocket or so, but this is pretty complicated, I remember faintly) - having a simple functional description with simpe antiderivative and then discuss this function. This would also make obsolete the existence of two different examples.
 * The only point which I would drop is the one on the Pythagorean theorem. Probably a reader willing to learn about the integral will know that the reals are a continuum. Even if not, when one starts out with a physical example, it should be intuitively clear to anyone that it is not enough to sum up the speed of the rocket at 1,2,3,4 seconds after take-off.
 * In the last subsubsection on Stokes, the f(x)dx is not what corresponds to the differential form &omega;, rather it is the antiderivative F(x). By mentioning above that F is called antiderivative, this can be included here without further ado. Jakob.scholbach 17:03, 7 July 2007 (UTC)

Request for comment
I and KSmrq are currently locked in dispute about the lead section of the article diff between preffered versions. As you can see, I've provided a precise source for my version. KSmrq has reverted it repeatedly without bothering to provide a counter-reference. On top of that, he has repeatedly engaged in incivil behaviour, such as calling me as a nuisance and lying to portray me in an unfavourable light. I request the community to look into the matter. Thank you. Loom91 16:43, 6 July 2007 (UTC)
 * Comments on the nature of this dispute can be found here, including some advice and policy reminders from mediators. Discussion pertaining to article content should remain on this page. &mdash; KieferSkunk (talk) &mdash; 01:29, 7 July 2007 (UTC)

Discussion
I can somewhat sympathise with both sides of this debate, but I've been trying to stay out of it because it has/is rather heated (for reasons that are beyond me -- it seems to me sensible compromises are available). The following are my own views on the various points Loom91 wishes to dispute.


 * 1) This seems a minor point, and can be quibbled about later. Refinement seems fine to me.
 * 2) I understand KSmrq's view that in referring to basic/simple integration it is continuous functions that are considered. However, it is adding an extra, and in practice unnecessary, qualifier to "function". The important issue is that of the continuity of the domain; I don't feel that referring specifically to continuous functions makes things at all easier to understand for naive readers. It's probably easier to leave it out.
 * 3) This point I'm already discussing. I understand the desire to remove the paragraph, and I am working to try and build a suitable replacement. I see no problems here presuming discussions do not get roadblocked for some reason, so this point is moot.
 * 4) dx does denote a weight, and can be read that way. Indeed, that was its use in Leibniz original notation -- it denoted an infinitesimal weight. Extending a little, with differential forms the basic forms do indeed denote densities which weight the various functions. There is some avoidance of this view in introductory texts because of the fear of infinitesimals (which it originally denoted) but I wouldn't say that it is wrong, especially given the differential forms, non-standard analysis, and smooth infinitesimal analysis approaches. Can we put both views on equal footing (rather than demoting one to being parenthetical, or removing it altogether) with something like "dx may be alternately viewed as either ... or ..."?
 * 5) Postponing tFToC till a proper discussion can be made, or promoting it to be more fully discussed earlier may be of benefit. This is more of a minor quibble as far as I can tell.

Hopefully this is helpful and can help cool the debate somewhat -- there are reasonable compromises here, so hopefully cool heads can prevail? -- Leland McInnes 18:45, 6 July 2007 (UTC)


 * I will like to point out that that was exactly what I had originally done, mentioning that dx can have different interpretations depending on the theory and giving three examples. KSmrq's version gave the impression that dx was treated as the variable of integration only in simplified texts for beginners, when it is a perfectly valid interpretation at all levels. Loom91 07:13, 7 July 2007 (UTC)

The phrase on what "dx" denotes seems to be the main conflict area now. It can most definitely be viewed as denoting a weight; it corresponds to the &Delta;i mentioned in the section on the Riemann integral. However, I think that most introductory calculus texts treat it as formal notation. The best way to treat it in this article depends on how the rest is written, but for the moment I agree with Leland's proposal so I implemented it. I removed the reference to the book by Dey because I can't find any information about it, suggesting that it is rather obscure. -- Jitse Niesen (talk) 19:37, 6 July 2007 (UTC)


 * I think treating it as a weight smudges the fine but important distinction between a Riemann sum and a Riemann integral. Loom91 07:13, 7 July 2007 (UTC)


 * The language of the intro is far too informal, anyway, so I don't see any reason to bicker over this small point. It's not the job of the intro to explain calculus to 4th graders.  For comparison, here is Britannica's entire article on "integral":
 * in mathematics, either a numerical value equal to the area under the graph of a function for some interval (definite integral) or a new function the derivative of which is the original function (indefinite integral). These two meanings are related by the fact that a definite integral of any function that can be integrated can be found using the indefinite integral and a corollary to the fundamental theorem of calculus. The definite integral (also called Riemann integral) of a function f(x) is denoted as


 * (integral[a,b] f(x)dx)


 * (see integration [for symbol]) and is equal to the area of the region bounded by the curve (if the function is positive between x = a and x = b) y = f(x), the x-axis, and the lines x = a and x = b. An indefinite integral, sometimes called an antiderivative, of a function f(x), denoted by


 * (integral f(x)dx)


 * is a function the derivative of which is f(x). Because the derivative of a constant is zero, the indefinite integral is not unique. The process of finding an indefinite integral is called integration.
 * Just my 2 cents. Gnixon 22:27, 6 July 2007 (UTC)


 * It's nice that Britannica is that concise, however I think the relevant points here are WP:NOTPAPER and WP:MTAA. We have entirely separate articles for Riemann integral, Lebesgue integral and Henstock-Kurzweil integral, as well as Antiderivative and more. Those articles can handle greater tehnicality as required (though we certainly delve into the technicality in this article). I think it important that, as a core article on calculus, this article be accessible to a general audience. Sure, the "formal definition" and later sections can provide detail that a general reader may not follow, but we should provide sufficient general material up front to provide an average interested reader with at least an understanding of the core concepts/ideas of integration. -- Leland McInnes 02:34, 7 July 2007 (UTC)


 * This is very important. The introduction needs to introduce; it is the most important part of the article for majority of readers coming to the page and needs to serve a non-mathematician reader (adult, not fourth-grader) as well as the mathematical.  It should not, for instance, use the (at that point) undefined term "Riemann integral". -R. S. Shaw 05:22, 7 July 2007 (UTC)

New lead
I have written a new lead, which summarizes the content of the article more closely, as recommended by WP:LEAD. I feel that the technicalities (continuous vs general functions, area vs signed area, description of the Riemann sum) should be defered to the main text. In particular, in the interest of clarity, I suggest replacing the present picture of a sign-changing function with a picture of a positive function. Arcfrk 06:14, 7 July 2007 (UTC)


 * I like the current lead. But while it discusses the Riemann integral in some detail, the general purpose of an integral, that of totalling, is not really alluded to, nor is its relation to a sum. I think the examples of various types of integrals should be replaced by some general discussion of why anybody wants to use an integral. Anyway, my purpose of initiating this RfC has suceeded: peace and sensibility has been restored. Thanks to everyone who helped out.Loom91 07:21, 7 July 2007 (UTC)


 * I agree with Loom91 that the current lede lacks an accessible description, which the previous lede offered. On the other hand I feel this is, perhaps, symptomatic of the current state of the "introduction" which I feel fails to provide the accessible description of integration that it should; were that fixed then the lede could suitably summarise that section and all would be well. I'll continue trying to discuss reform of the "introduction" section and hope this will clarify lede issues. -- Leland McInnes 14:18, 7 July 2007 (UTC)


 * As pointed out previously by Leland McInnes, part of the difficulty here is the sprawling nature of this article. It would certainly be nice to have a one or two-sentence description of integration as a continuous version of summation. On the other hand, we should try to avoid focusing too much on details of the Riemann integral, since it is treated in a separate article, with its own lead, introduction, and explanations. I was not able to come up with anything that at the same time adequately reflected the idea of integration and were concise enough and gentle enough to warrant inclusion in the lead. If anyone can accomplish it, please, do! Arcfrk 19:46, 7 July 2007 (UTC)

Recent edits
I think the text flowed better before the recent edit by Loom91. Comments? Oleg Alexandrov (talk) 15:20, 9 July 2007 (UTC)


 * There are a lot of problems with it, especially with the lead. I suggest reverting back. Arcfrk 16:18, 9 July 2007 (UTC)
 * I have a number of problems with the recently added section. Whether nature is essentially continuous or not (something which is open to debate) has less to do with the integral than the purpose of its mathematical formulation: finding areas under curves. I think (a rewrite of) this added paragraph could possibly be transferred to a section on the scientific impact of the integral, but I do not think it needs to go in the leading statements. Xantharius 16:50, 9 July 2007 (UTC)


 * I believe that Loom91's edit was in good faith, but I have reverted back (preserving Xantharius's later contribution) since the newly introduced problems are far too severe and numerous to fix, and with continuous editing of other sections, the longer we wait, the harder it becomes to revert. Arcfrk 19:03, 9 July 2007 (UTC)

I would have liked it if you gave reasons why you consider all of the changes I made in that edit to be detrimental when reverting. It seems the main objection is with my changes to the lead and the ordering of the sections, so I will try restoring the rest of the changes I made. If those are reverted also, I'll like to know what was wrong instead of simply "introduced problems are far too severe and numerous to fix". For now, let's discuss the lead and the ordering:


 * It seems very strange to begin an article with history. Only readers with specialised needs will want to learn the history of mathematical concepts when they come to Wikipedia, the majority will want to learn the concepts. See, for example, Manifold. The convention is to write technical articles in the order of difficulty/specialisation, with more accessible sections or sections of more general interest coming first.
 * The lead is still lacking any sort of introduction. A reader who does not about integrals will be bewildered after reading the intro, having no idea what an integral does (apart from the very specific example of finding areas) or why anyone needs one. I tried to provide motivation and an intuitive idea of the integral as something to take the concept of totaling (the motivation behind summation) into continuous domains, whose usefulness comes from the continuity of Nature. In all current theories of physics, including the Standard Model/GTR (the currently accepted theory of physics) space-time is continuous. Discreet theories of space-time are still over the horizon. Loom91 06:54, 10 July 2007 (UTC)

I still hold that the lede will remain contentious until the article is given a more stable form. They are perhaps the most difficult part of the article to write, so great care needs to be taken when putting one together. I would prefer that all of this energy be directed into refining the newer sections and improving general flow of the article.--Cronholm144 07:43, 10 July 2007 (UTC)


 * Loom91 is again being destructive, not helping. The latest edit gutted the massive amounts of work put into the numerical quadrature section. (I reverted, of course.) This lack of judgment has really got to stop. --KSmrqT 11:03, 10 July 2007 (UTC)


 * I disagree that Loom91 is being deliberately destructive. As Cronholm pointed out, the lead may be a point of contention (and therefore should probably be left alone for now) until the rest of the article is stabilized - the lead should ultimately summarize the rest of the article in 1-4 paragraphs (it should be a nutshell).  Also, please remember to assume good faith unless it is clear that another user is deliberately vandalizing or harming the content.  Having looked over the history here, I do not see that this assertion is clear here. &mdash; KieferSkunk (talk) &mdash; 18:29, 10 July 2007 (UTC)


 * While I agree that Loom91 may not be deliberately destructive, he is certainly being accidentally destructive. That's how I read KSmrq's "lack of judgment" remark.  Presumption of good faith is not an argument to allow this sort of behavior to continue.  And, as long as people like KSmrq, KingBee, and others are working on making substantive and much appreciated contributions to the article, nuisance refactoring edits like those of Loom91 are going to continue to be disruptive.  Silly rabbit 18:36, 10 July 2007 (UTC)


 * (nod) I'm not advocating letting the behavior continue - if you do feel that the edits are not helping, then by all means, undo and discuss. But a request for comment in WP:WQA was filed mostly because of the way KSmrq was addressing Loom91 in this dispute, and that's what I'm addressing with WP:AGF.  It's important for EVERYONE to keep discussions in the article talk page focused on the article content, and to keep statements of a person's character, personal attacks, "snide comments", etc. out of the discussion as much as possible.  Loom91 has requested comment further up in this thread about what was wrong with his edits - please respond to those points with specific regard to the content. &mdash; KieferSkunk (talk) &mdash; 18:56, 10 July 2007 (UTC)


 * BTW, if the dispute continues and you are unable to reach a consensus on the content, please consider filing a Request For Comment, which will bring the dispute to the attention of more editors. Sometimes getting an outside view from uninvolved editors is the best way to resolve such a dispute.  (I would be happy to provide my own analysis and commentary on the content, but as I pointed out before, I am not qualified to do so because I don't have much personal knowledge in this area.) &mdash; KieferSkunk (talk) &mdash; 19:00, 10 July 2007 (UTC)


 * Hello, again, KieferSkunk. I've been rather busy, but pardon my manners for not thanking you before for your effort to look in on Loom91's complaint.
 * May I suggest that you read the section of this talk page entitled "The lead" from start to finish. You will notice that it begins with Loom91 dramatically asserting "KSmrq, I'm afraid your edits are simply incorrect." By the end, each of his five points has been rebutted, thoroughly and repeatedly. (I finally invested some of my time to do what he would not, despite my urging; I quoted exactly how some respected authors treated continuity, as I had said from the beginning.) And I was merely one more in an unending stream of editors to rebuff Loom91.
 * Then I invite you to notice that, during the month of June, I was a major contributor to both the talk page and the article, discussing numerous subtle issues and technical details, as well as matters of presentation. You will also find that I am a regular hand-holder at the Mathematics Reference Desk. Nor am I alone; we have editors with a range of knowledge and tastes all having problems with Loom91's judgment and behavior.
 * The problem here is not one of our etiquette, nor of whether to assume good faith, but of repeated bad claims and bad edits by Loom91. It is not acceptable for him to tie up the time of other editors by an endless series of mistakes, then expect that we will each time patiently revert and explain. And you will find that he has a history of such behavior, some of which I have linked in the thread I mention.
 * His bad behavior drives away good editors. King Bee was creating a much-needed new section on complex integrals and residue calculus, complete with illustrations; Loom91's misbehavior has repelled him. This is unacceptable.
 * I have taken a little of my time to discuss with you issues raised by Loom91's conduct. Once again, it is time that I would prefer to have spent filling out and polishing the content of the article (which still has far to go). But since you took the time to look in, I feel you deserve the courtesy of a comment. Thanks again for your concern. --KSmrqT 03:26, 11 July 2007 (UTC)


 * I understand where you're coming from, KSmrq. I've been in similar situations myself.  I can also understand how you would get frustrated in that situation, but I must emphasize that it's important to remain civil about your disagreements.  As I said, I'm not advocating letting the disruption continue - instead, filing a Request For Comment can help get more eyes on the matter and help everyone reach consensus.  If, after reaching a consensus, the editor in question continues to go against it, you can use any of the dispute resolution techniques available.  There are also some notes on WP:CON for how to deal with people who reject or ignore consensus.  I hope these will help. &mdash; KieferSkunk (talk) &mdash; 18:42, 11 July 2007 (UTC)

General properties
I started editing the general properties section, and found that the functions which were being integrated were mapping into a field. I do not think this is true: how is one meant to multiply elements of, say, the finite field Z7 by elements of R in order to do, for example, Riemann integration? This section needs more careful thinking about, and I suggest moving it to after the discussion of Lebesgue integration for the following reasons:


 * If we want the linear functionality to be discussed in its most general setting, and for the resulting set of integrable functions to be a vector space, then the functions to be integrated all must be defined on at least a measure space, and the same measure space at that.


 * The co-domain of the functions to be integrated must be compatible with multiplication by real numbers, since measures map into the real numbers in general.


 * The definitions of Riemann and Lebesgue integration are more important than the general properties, especially since it's almost impossible to state in generality the properties without a discussion of measures.

I think that this means that the functions need to be real- or complex-valued. I've never seen integration defined for other types of function, but I'm willing to hear it before making any changes. And if we want the general properties to indeed be general, we'll have to have measures defined first. Xantharius 20:16, 10 July 2007 (UTC)


 * In fact, since this was removed from a previous version anyway, I propose that general properties be moved to just prior to Basic inequalities as it will then follow the mention of all of the other types of integration, but be placed before we start discussing real-valued functions. Xantharius 20:21, 10 July 2007 (UTC)

Move and correct it.--Cronholm144 20:38, 10 July 2007 (UTC)


 * Done. Xantharius 16:42, 11 July 2007 (UTC)

The current state of the General properties section seems to defeat its main purpose, to provide a picture of the completely generalised integral. Confining the discussion to R/C and using measures does not seem to serve this purpose. Measure theory is not an absolute necessity to work with integrals (Daniell integral, for example) and all integrals do not map to the real or the complex numbers. Integrals may also be vectors. That is why I wrote that the inequalities are valid only when the domain of the functions is a metric space. I had tried to point out features that any reasonable definition of integral must have. Having the real numbers as the domain does not seem to fit htis criteria. I request that the section be made free from explicit references to measure and the exact domain of the functional. In fact, I'm not sure there is a technical necessity for it to be a field. Loom91 21:26, 11 July 2007 (UTC)


 * At the very least the domain has to be continuous, or else you're doing ordinary summation, not integration. -- Leland McInnes 22:27, 11 July 2007 (UTC)


 * Domain. I agree that the Daniell integral can be defined on a general set X, but all of the functions to be integrated must have the same domain. If I have an integrable function defined on [0, 1], it is not meaningful to add that to an integrable function defined on [2, 3], so this must be specified.


 * Co-domain. As far as I can tell, the current defintions of Daniell integral, Henstock-Kurzweil integral, Lebesgue-Stieltjes integral and Riemann-Stieltjes integral on Wikipedia have the functions being real-valued (or complex-valued), not mapping into some general set. At the very least there has to be some compatibility between the values produced by the function, and scalars to indicate "length" of some sort. I would want to be precise about this if we are going to talk about it at all.


 * I, too, would like to have this made as general as possible. We can go back to something more general if there is good formulation, but, unless I'm going to be very surprised, the co-domain will have to be something more specialized than a general set. As an example of this, someone needs to explain to be how I perform integration if the co-domain is the long line. Xantharius 03:18, 12 July 2007 (UTC)


 * Well we can probably pin down minimal requirements. I think the primary points of concern are certain topological properties, and certain algebraic properties. On the algebraic front, I think it is reasonable to ask for a field, and leave that as the requirement. On the topological front I suspect we require non-discreteness (singletons are not open) and completeness. That essentially means we want a topologically complete field. The most immediate example of such a thing that springs to mind for me that isn't R or C would be Qp, and interestingly enough integrals for Qp exist using the Haar measure. If someone has an actual source for the required conditions, as opposed to my basic speculations, please feel free to raise it. In the meantime I think the p-adic case is enough to show we can go beyond R and C. -- Leland McInnes 05:14, 12 July 2007 (UTC)


 * I think summations can be treated as a special case of integrals. There are several ways to do this. Apostol uses step functions with ordinary Riemann integrals, a book I don't remember the author of used Riemann-Steiltjes integrals, and I suspect it can also be done using Dirac delta distributions. I will also like to point out that the codomain of the functions and the codomain of the integral need not be the same. Vectors may be integrated to give scalars. Moreover, the codomain of the integral need not be a field, since integration of vectors may give either scalars or vectors. Loom91 07:36, 12 July 2007 (UTC)


 * I've been letting others work on the general properties section, but I do have a comment. It is a strategic mistake to try to find the most general possible language to cover every use of every integral. Why?
 * Because it is virtually impossible.
 * Because much of interest cannot be said in such generality.
 * And, most importantly, because to do so requires technical machinery and language that our typical reader will not understand to achieve results of little benefit.
 * Already it is awkward to precisely state linearity, since that requires a vector space of functions — not a trivial concept to introduce. (But this one is important.) We must discuss the Fundamental Theorem, though different integrals place different constraints. We should make a few powerful statements about the most important examples, especially real and complex integrals of the Riemann and Lebesgue kind. We can indicate important differences between these cases, and let the reader know there is more to learn.
 * It is quite wrong about this section that "its main purpose [is] to provide a picture of the completely generalised integral." We rarely attempt that in specialized articles, much less a springboard article like this. Mea culpa; I suppose I should have spoken up earlier. --KSmrqT 12:03, 16 July 2007 (UTC)
 * Perhaps not every possible integral, but vector valued integrals and Daniell integrals are both fairly well-known, so the current emphasis on measure theory and real/complex valued integrals seems misplaced. As Leland says, restriction to only these special cases takes away the value of the section. As for your concern of complexity, wiki is not paper and we must cater to the very specialised as well as the general audience. In any case, the current approach using measures is more technical and jargony than my version, which used only some basic concepts. I think the consensus is with me on this edit, just read this thread. Loom91 12:28, 16 July 2007 (UTC)
 * What exactly are you proposing? In one of your edits, you changed real/complex numbers to a field.  This clearly fails on topological grounds.  You can define integration of functions into a broad class of topological vector spaces, if you want to go down that route.  But in fact, the functions need not be into a field, and their codomain must have some kind of nice topology.  Silly rabbit 13:21, 16 July 2007 (UTC)
 * Firstly, I distinguished between the range of the integral and the range of the functions, they may not be the same. I defined the range of the integral as a linear space. This seemes to be well behaved enough, since we are not really doing any operation on the integrals themselves. As for the range of the functions, how will linearity work meaningfully if it is not at least a field? As for additional topological properties, that may be necessary, though I can't find a compelling reason. Xantherius believes that the domain can be any general set, though I don't think how we can define linearity in that case. In any case, I think everyone can agree that limiting the discussion to measure theory is too restrictive? Loom91 13:43, 16 July 2007 (UTC)
 * The section looks a bit schizophrenic. On the one hand, you allow that functions can have values in a vector space.  On the other hand, you require that the integral takes values in a field.  If the functions are vector-valued, then the integral should also be vector-valued.  (Or at least, we should not exclude the possibility that the integral produces a vector instead of a scalar.)  Silly rabbit 14:04, 16 July 2007 (UTC)
 * You have got it the wrong way around. I wrote the opposite: I required the integral to map to a linear space (usually an Euclidean n-space) and the functions to map to a field, though on hindsight the latter is too restrictive. We cal also have vector valued functions. In that case, we can make both the ranges linear spaces and when stating linearity use as scalars the elements of the underlying field of the range of the functions. How does that sound? It's also purely algebraic, with no reference to topology. Loom91 14:26, 16 July 2007 (UTC)
 * No, Xantharius does not believe that the domain or co-domain can be any general set, contrary to Loom91's assertion that he (I) does (do).
 * While the current version of General properties is (possibly) more technical, it is at least correct. Topological properties are necessary because integration, however you define it, is a limiting process, and limits require topology in order to be defined.
 * Loom91 keeps asserting that the range of the function and the range of the integral can be different, but this, as far as I can tell, is wrong: if you sum up images of vector-valued functions, you get a vector; if you sum up images of scalar-valued functions, you get a scalar. If someone has a counter-example, either put me out of my ignorance by leaving it on my talk page or here.
 * I checked a reference (Royden), and Daniell integration has the functions mapping from a general set X into R (which was needed because suprema and infima were required for the definition).
 * The essential property wanting to be defined for this section is that the set of integrable functions, however you want to define them, is a vector-space. My proposal: have the functions f : X &rarr; V, where V is a topological K-vector space for a field K, and I : L &rarr; V, where L is defined to be the set of all integrable functions on X. We can get rid of the measures if we must, and then specify for the Riemann and Lebesgue integrals as examples. Then integration is shown to be linear however you define it, and we still get basic cases as examples. If there is no dissent, I'll go ahead and do this so we can move onto something else! Xantharius 17:20, 16 July 2007 (UTC)
 * Supplementary comment after reading KSmrq's comments in more detail: Define vector spaces of functions in brief, discuss the property of linearity for the Riemann and Lebesgue integrals next, and then generalize as outlined above. Xantharius 17:24, 16 July 2007 (UTC)

You yourself had claimed above that the Daniell integral could be defined for an arbitrary set X, I did not make that up! Anyway, I gave the counter-examples above. The surface integral of a vector is a scalar. Further, the integral of a vector may be a vector. Does anybody contest these elementary examples? For now, getting rid of measures will be a start. And I suggest that since Riemann and Lebesgue integrals are already sicussed in detail before, those examples come after the general discussion. Loom91 21:25, 18 July 2007 (UTC)


 * In the way this article is presently structured, and I believe it to be prudent, surface integrals appear as an extension of the concept of integration. Therefore, all the properties of integral in the "General properties" section can only refer to Riemann or Lebesgue integrals of scalar (real or complex valued) functions which have been discussed at that point. It is an abuse of languge to refer to properties of "integration of differential forms" as "properties of integral" in any case.
 * Ksmrq explained elsewhere on this page why trying to state the most general properties that cover all possible types of "integrals" is a BAD IDEA. Arcfrk 22:08, 18 July 2007 (UTC)


 * I don't understand your point. Surface integrals or integrals of differential forms are not integrals? Also, if we are going to limit ourselves to a discussion of the properties of Riemann and Lebesgue integrals, then why have the section General propoerties? Loom91 11:29, 19 July 2007 (UTC)
 * I also wonder. It seems to be too difficult to state even the linearity precisely.  Even with a precise statement available, it would probably be of rather limited usefulness to anyone.  Those who are capable of reading it would already know about it, and the others would not get anything at all out of it. Silly rabbit 12:00, 19 July 2007 (UTC)

Lp spaces
Should we include a short introduction to this which leads to the Lp space article? Xantharius 16:53, 11 July 2007 (UTC)


 * Yes, I suggest a remark on Schwarz inequality at the "Basic inequalities" section. together with a brief remark on Lp spaces. Jakob.scholbach 20:35, 18 July 2007 (UTC)

The length of the Numerical Quadrature section
I realise that a lot of effort has gone into writing that beautiful section, but surely the length and the level of detail is disproportionate for a broad overview article? Numerical evaluation of integrals is only a part of the theory of integration. To discuss it using actual worked examples and then proceed to point out various very technical difficulties and how to resolve them seems an unnecessary level of detail. In my edit I had tried to preserve essential content while commenting out topics of secondary interest. Isn't this the standard approach for overview articles? I will also prefer if the discussion was about content rather than editors and alleged ulterior motives. Loom91 21:35, 11 July 2007 (UTC)


 * Umm... Loom your recent edit that you cited as per talk looks very similar to the edit you made earlier as part of an edit war. I am not sure what you stand to gain from doing this, but it is likely to incite another reaction.--Cronholm144 09:30, 16 July 2007 (UTC)


 * I was forced to make the same edit again because all other editors seem uninterested in discussing changes they don't like unless someone actually makes them. Have you noticed how long my comment has stood without any replies? Such a long section containg information of highly specialised information is inappropriate for an overview article like this (as pointed out by MoS). KSmrq asked me to discuss changes before Imake them, but since he has not bothered to take part in the discussion I was left with no other choice. Loom91 09:38, 16 July 2007 (UTC)


 * Why don't you contact him directly and nicely on his talkpage? There is always a choice that doesn't involve a continuing slow moving war.--Cronholm144 09:46, 16 July 2007 (UTC)


 * Loom91, no one "forced" you to make any edit.
 * Your edit summary does not mention that you are doing this again. That is deceptive, and troubling.
 * And in my part of the world I have a life and a weekend. And this material has been discussed previously on this page if you would bother to look. And the table is, in part, a "figure" used to help explain Romberg integration. And no one has chimed in to agree with you. And as repeatedly pointed out, this article needs new and expanded sections, not cuts to valuable material. And on, and on, and on, and on.
 * I have had many productive discussions here, with informed people who are interested in listening, cooperating, and contributing productively. Discussions with you do not seem to fall into that category. --KSmrqT 11:30, 16 July 2007 (UTC)

Harvard citation style
There is the template Template:Harvard citation no brackets,which does not display the brackets around a citation. When several references are shown in a row, I'd propose to use this instead of the format used right now, because something like "..... (author1, year1)(author2, year2)...." looks rather odd. Jakob.scholbach 03:08, 13 July 2007 (UTC)


 * Style guides say to use one pair of parentheses, with semicolons separating the sources. It's easy to do it; the Harvnb template works. For example,
 * Studies of telephone directories have shown that some names are far more common than others.
 * The wiki markup for this example is
 *  Studies of telephone directories have shown that some names are far more common than others . 
 * As usual, an automatic link to the entry (which is listed in alphabetic order) is created. --KSmrqT 06:45, 13 July 2007 (UTC)
 *  Studies of telephone directories have shown that some names are far more common than others . </tt>
 * As usual, an automatic link to the entry (which is listed in alphabetic order) is created. --KSmrqT 06:45, 13 July 2007 (UTC)

Recent changes to content
Focusing on content a bit, rather than the revert war going on, looking at this edit it appears to me that KSmrq's text is better written and more clear. Comments? Oleg Alexandrov (talk) 15:47, 16 July 2007 (UTC)


 * Yes, I agree with you.
 * I prefer "Integration is essentially a linear functional, (KMsrq) over " Integrals are essentially linear functionals" (Loom91). Integrals themselves are not considered to be a functional, rather the map
 * $$f \mapsto \int f dx$$
 * is the functional.


 * The formulation "If both E and F are ordered, then the following set of inequalities are also conventionally assumed to hold" (Loom91) is little helpful, because it is unclear what "conventional assumptions" are.
 * KMsqr's version is much clearer. So, "Let L be the set of all functions for which we are defining integral (the integrable functions) and F be a linear space . All functions in L must have the same range and the same domain for consistency. Let the range of the functions be the field E, which may or may not be identical to F." (Loom91) is just misleading, not explaining what the linear space F should be. Jakob.scholbach 17:21, 16 July 2007 (UTC)


 * I will go ahead and incorporate these into the changes I proposed near the bottom of General properties unless there are any objections. Xantharius 18:04, 17 July 2007 (UTC)
 * Go ahead! Jakob.scholbach 22:16, 17 July 2007 (UTC)
 * I have executed the recent proposed changes (see above) as regards linearity of the integral. In addition, I expanded some of the other properties in there, added a section header for this (Properties) and adjusted some of the Conventions section which, unfortunately, is now again in the Properties section (but I do not think either of these things belongs in a section for the definition of the integral as they're not definitions but consequences of those definitions). I have concentrated on Riemann integration for most of this, while mentioning Lebesgue integration, and still having the important generalized properties for any integral as a subsequent part of the discussion. I hope this gets us a little closer to where we wanted this section to be. Xantharius 19:37, 18 July 2007 (UTC)
 * The sentence "It is necessary, for the integral of a function f to be defined on a closed and bounded interval [a, b], that f be bounded on that interval" is questionable, if there are discontinuities at a measure-0-set (for Lebesgues-integration) (e.g. having value f(n)=1/n, n a natural number, f(x)=0 elsewhere (in particular also at x=0). (Actually people with less rigorous understanding of the term "function on [a,b]" (i.e. not assuming that it needs to be defined everywhere) may also think of improper integrals, which are also linear, if defined at all). Why not just say "Given an interval or more generally any measurable space, the set of functions integrable on that set form a vector space, i.e. ..."?
 * I'd merge the inequality about $$\int f(x) \leq \int g(x)$$ with the first one, which is just a special case. At the conventions section (when talking about $$\int_a^b vs. \int_b^a$$) a short remark leading to the orientation of manifolds may be in order.Jakob.scholbach 20:20, 18 July 2007 (UTC)
 * I think it is not correct to allow integrals having values in any topological K-vector space, K a field. (Think of K=Q, where you could not do the limiting process). Rather I guess, one needs a (topologically) complete field K and a vector space over this K. Jakob.scholbach 20:34, 18 July 2007 (UTC)
 * I'm sorry: you're quite right about the "complete" part, which I forgot to put in. As for the other stuff about boundedness, the article has been changed since the remarks you've written. I'll have another look. Xantharius 20:40, 18 July 2007 (UTC)
 * Completeness: done. As for the other stuff, I've been working on this for hours today to get the balance right between explanation and correctness, and I'm done for now, but feel free. I think it would be a shame to lose the information from the inequalities you mention for two reasons: each is found in plenty of texts, and a lot of people don't realize, for example, that m(b &minus; a) is a function. I'll do the measure zero stuff right now. The vector space stuff is mentioned at the end of the general linearity section, and I left it there so the initial discussion wouldn't be cluttered, but do what you think is best. As for the manifolds, I'll leave that to you!Xantharius 20:55, 18 July 2007 (UTC)
 * Actually, for Riemann-integrable functions (which are the focus of that section) don't the functions have to be bounded? Xantharius 20:55, 18 July 2007 (UTC)
 * Yes, if the function is supposed to be defined everywhere in the closed interval, then it is bounded there automatically, but I think if two functions f and g are defined at the interval minus some (finitely many) points and both (improper) integrals of f and g are defined (ie. converge), then so does the integral of (f+g). Jakob.scholbach 04:37, 19 July 2007 (UTC)
 * I concur. I think we might just leave it as it is currently, though: this might be getting too technical for the scope of the article. Xantharius 10:07, 19 July 2007 (UTC)

polished introduction
I boldly polished the introduction a little bit, hoping that the epsilon I added is positive (possibly spoiled by a negative delta given by linguistic flaws). My intention was to point out more clearly the counterpoint between adding finitely many function values and infinitesimal steps. I deleted the excursion on the Pythagorean theorem, which seemed (to me) to lead a little bit too far, but the idea "infinitely fine" should still be there, I hope. (I admit that the line summing all the square-roots of n/5 looks a little ugly. Perhaps it would look better in mode).

Should this section be moved to the first place? (I find it much more important than History and terminology/notation).

Jakob.scholbach 21:54, 18 July 2007 (UTC)


 * Looks like good improvements to me. Hopefully when I can find some time (I've been very busy the last week, and will be busy for at least the next week or so moving) I will try to make some improvements of my own. -- Leland McInnes 02:38, 19 July 2007 (UTC)


 * Introduction should normally follow the lead. However, I felt that at present, the history section provided a better introduction to the subject than the section titled "Introduction"! Arcfrk 03:15, 19 July 2007 (UTC)


 * I'll explain why I think this is not the most helpful way to go, but for training purposes, here's the summation as written
 * &radic;1&frasl;5*( 1&frasl;5-0)+&radic;2&frasl;5*( 2&frasl;5-1&frasl;5)+...+&radic;5&frasl;5*( 5&frasl;5-4&frasl;5) ≈ 0.7497
 * here it is with a facelift
 * &radic;1&frasl;5×(1&frasl;5−0) + &radic;2&frasl;5×(2&frasl;5−1&frasl;5) + ⋯ + &radic;5&frasl;5×(5&frasl;5−4&frasl;5) ≈ 0.7497
 * and here's the TeX display version
 * $$ \sqrt{\frac15}\times\left(\frac15-0\right) + \sqrt{\frac25}\times\left(\frac25-\frac15\right) + \cdots + \sqrt{\frac55}\times\left(\frac55-\frac45\right) \approx 0.7497 . $$
 * The facelift version would be used inline, so let me highlight the fine points.
 * An ellipsis should never use three dots, but a single character; and between plus signs that should be the centered version.
 * Never use a hyphen when a minus sign is intended; use the "<tt>&amp;minus;</tt>" character.
 * The asterisk character denotes multiplication in programming languages, but in mathematics we use either a centered dot ("<tt>&amp;sdot;</tt>") or a centered times sign ("<tt>&amp;times;</tt>") (not a literal "x") or nothing. On the rare occasions when we want a centered asterisk operator, we use "<tt>&amp;lowast;</tt>" ("&lowast;").
 * Spacing is an art, and inline we must sometimes use no-break space ("<tt>&amp;nbsp;</tt>").
 * Note that the "Insert" line below the edit window provides most of these special characters. (The "minus" is the one next to the "times"; the first two characters are an "en dash" and an "em dash", and are intended for other purposes.)


 * Perhaps it's time I tried to explain what I was aiming for (but perhaps missed), beginning with the second paragraph ("Measuring depends on counting, …").
 * I began with the idea introduced by Leland McInnes: "Integrals generalize discrete sums to continuous sums." Indeed. However, from experience, I felt that lay readers would be troubled by "continuous sum", and by "limit".
 * Now I realize that I'm swimming upstream by not saying limit, because we (the initiated) are so conditioned to that language. So, I wrote two parallel paragraphs.
 * In the first, I talked about &radic;2, and how that very natural length could only be approximated by rational numbers; it required a new approach to number, beyond counting.
 * In the second, I talked about the area under the curve y = &radic;x, and how that could only be approximated by rectangles; it, too, required a new approach.
 * Just as I introduced no explicit construction for real numbers (no Cauchy sequences, no Dedekind cuts), I introduced no explicit construction for integrals. We are perfectly happy to use &radic;2 as a thing in itself, and the fact is that we routinely use integrals with absolutely no reference to limits. In fact, for many decades great mathematicians like Euler had no such limit construction to work with. And non-standard analysis made it clear that integrals are not limits and do not require them, no more than real numbers "are" Cauchy sequences and can be described in no other way.
 * As I said, I know I'm swimming upstream; but I think it is absolutely the right thing to do. I'm trying to talk about what is an integral as separate from one way we might define it.
 * That is one reason why my next paragraph went straight to the fundamental theorem, not to limits.
 * And when I do talk about definitions, I try to treat them on an equal footing. This is desirable for several reasons. Please try to free your mind from a "limits only" view of the topic. It is historically wrong; it is pedagogically wrong; it is factually wrong. For example, Rudin, Real and Complex Analysis, p. 19, defines a Lebesgue integral as a supremum over all integrals of simple functions everywhere less than the function of interest; no limits are used, not even to define measure.
 * The "vector lattice" (Riesz space) approach to integration theory suggests that orders are more important than limits. (In fact, I've been meaning to suggest that the approach to properties could benefit from that technology. But I have other things to say about that section!)
 * When I look at the latest revision, I believe two important things have been lost.
 * The discussion no longer refers to common, tangible experience.
 * Our focus on the concept has been replaced by too much emphasis on the calculation.
 * I know Leland was not entire satisfied with my rewrite; neither was I. Frankly, I'm not convinced my Pythagorean example was tangible and familiar enough, nor am I convinced my attempt at insight by analogy was clear. Small wonder that Jakob, perhaps not appreciating its hoped-for role, removed it as clutter.
 * I close with a meta-remark. Compare the length of this talk entry with the length of the introduction I wrote. And, I have not said everything I might about my thinking. When I said it would be easier to show rather than tell, this is what I meant! --KSmrqT 14:33, 19 July 2007 (UTC)


 * Unfortnately I don't have time at the moment to go into real detail, but this is a topic I feel is important, so I'll try to get the ideas and opinions I feel are most important across while it is under discussion. I feel the characterisation as a "continuous sum" is important, but I feel the current approach to this is not ideal. While it is true that we can characterise the difficulties of the continuous in terms of the real number system and incommensurability, I believe going this route with discussion is both unnecessarily technical, and diverting from the main topic. I think the continuous and the discrete are concepts that are sufficiently intuitive in their own right that, with a little prompting via examples, we can use them whole and not resort to describing the technical features of how mathematcians interpret and work with them. That is, one does not need to know or understand anything about limits, real numbers, or the technical details of incommensurability, to appreciate the distinction between continuous and discrete, and ultimately it is this distinction that describes what integrals are; the rest is technical icing on the cake which can be more appropriately dealt with in the explicitly technical sections.
 * I guess, in a sense, I am wishing to swim even further upstream than KSmrq for the introduction and remove not only limits (which I agree, are not technically required in a variety of formulations) but also specific notions of continuity via the real numbers (which are also not required if we were to work with, say, smooth infinitesimal analysis). To express, at core, what integration is requires only the intuitive notion of continuity; how the continuum is interpreted or understood should not be required, and by leaving it out we make the introduction far more accessible. Leland McInnes 16:01, 19 July 2007 (UTC)


 * Thanks for your comments! It is true that the emphasis is more on calculation than before. I was driven by my belief that to see the differences x_{i+1}-x_i becoming smaller and eventually getting replaced by differentials instead of differences is the key point. I remember myself staring at
 * $$\int f(x) dx$$
 * as a kid, wondering, what this notation means. Devoting some space to write down one calculation of an approximation was intended to help such a kid like I was. Insofar I would not concur with you, KSmrq, saying "we routinely use integrals with absolutely no reference to limits". This is certainly true for everybody knowing integrals and also for those kids who are just taught: integrating x^p, sin(x) etc. means semantically replacing it by blablabla. But overcoming the very step from the discrete to the continuous is, I'm sure, something which is very non-routine to every learner once in his education. Otherwise, why did it take so long to arrive at this notion? So I question your saying "Pedagocically it is not correct to use limits).
 * As for "Factually it is not correct to employ a limit": Lebesgue-integration is not defined by a limit, but a supremum-procedure. But, taking a supremum of a set of functions is a limit, namely the direct limit of the (ordered) direct systems of these functions. I certainly don't want to write this in the introduction :-), but I do think that some form of limiting process is inherent to both Riemann and Lebesgues integration (I don't know this Riesz-space approach and the article on it was not very informative, so I can't give a qualified statement on this right now). In any case, here I don't see that much of a difference between your, KSmrq's, version and my one: You wrote: 1st: 1-step approx, 2nd: 5 steps, 3rd: 12 steps and "As before, using more steps produces a closer approximation, but will never be exact. Thus the integral is born" - I was essentially just replacing your "Thus" by the (more explicit, I hope and pinning down more clearly) "The key idea is the transition from adding finitely many differences of approximation points multiplied by their respective function values to using infinitely fine, or infinitesimal steps."
 * Last, but not least "Historically it's not correct to explain it using limits": I agree that historically the limit process was not the first step. Perhaps Newton and Leibniz were just (of course genially) applying semantic rules on how to integrate(?). But, we are not writing the introduction to an article "History of integration". Therefore I think it's (for example, striving for an intro as short as reasonable) OK to skip the early days of integration, arriving directly at our modern understanding of integrals.
 * If you two or anybody else is able to give the reader a clear feeling what an integral is without mentioning the delicate transition "discrete vs. continuous" and without mentioning and implying without mentioning any kind of limit, I'll more than curious to know (and read it in this section!). Cheers Jakob.scholbach 17:35, 19 July 2007 (UTC)


 * Just to be clear: I believe that the delicate transition from the discrete to the continuous is the key point, I am simply suggesting that we can and should discuss that without resorting to the technical details of how that specific transition is managed: that is the job for the formal definitions. There seems to me to be too much focus on how to calculate integrals rather than explaining what integrals are -- at least as far as is required for the introduction which I think should be informal and accessible. -- Leland McInnes 18:24, 19 July 2007 (UTC)

The range of the functions and the range of the integral match
. . . unless I am very much mistaken. I think I know what Loom91 means by "the surface integral of a vector is a scalar." While this is imprecisely stated, taking, say, a line integral of a function F(x, y, z) into R3 over a closed curve C:
 * $$W = \int_C \mathbf{F} \cdot d\mathbf{r}$$

is indeed a scalar. But that's because F &middot; d&thinsp;r is a scalar, since F &middot; d&thinsp;r is the dot product of F with d&thinsp;r, that is,
 * $$ \mathbf{F} \cdot d\mathbf{r} = \left( F_1(\mathbf{r}), F_2(\mathbf{r}), F_3(\mathbf{r}) \right) \cdot \left( \frac{dx}{ds}, \frac{dy}{ds}, \frac{dz}{ds} \right)$$

whence
 * $$W = \int_C F_1 \,dx + F_2 \,dy + F_3 \,dz$$

where the Fi are the projections of F. And indeed, this is a scalar, because it's the sum of three functions which themselves are scalar-valued.

But, instead, consider F(x, y, z) = (0, 0, 0) for every (x, y, z) in X = R3. Then, over closed subset C of X we have
 * $$\int_C \mathbf{F}(x, y, z) \, dV = (0, 0, 0)$$

which is a vector. The last of these is an example of where the range of the function and the range of the integral match. But the first, the line integral, is not a counter-example of the range of the function and the range of the integral not matching, because you are not taking a straight integral of the vector-valued function F: you're messing about with it before you integrate, and what you are integrating is not, I repeat, not a vector-valued function: the expression F &middot; d&thinsp;r is a scalar. I stand by above comments: the range of the function and the range of the integral have to match. Xantharius 02:14, 19 July 2007 (UTC)


 * Thank you for posting a detailed explanation, you are quite correct, and I have already briefly commented on it above. On the other hand, it's worth keeping in mind that the talk page of the article is primarily intended for discussing improvements to the text, not for dispelling anyone's confusions and teaching them the subject so that they might edit the article! Arcfrk 03:10, 19 July 2007 (UTC)


 * You are, of course, completely correct: I should probably have put it on my own talk page or something. Nevertheless, I think the point has been made! Thanks for the constructive comments. Xantharius 10:02, 19 July 2007 (UTC)

axiomatic characterisation of integration?
Is there an axiomatic characterisation of integration? If so, this may be worth a word (around the general properties section). Jakob.scholbach 05:11, 19 July 2007 (UTC)
 * You might want to have a look at Daniell integral if you haven't done so already. There might be enough there to draw upon for what you're proposing. Xantharius 10:04, 19 July 2007 (UTC)
 * I can provide the originals by Daniell if you would like.--Cronholm144 10:28, 19 July 2007 (UTC)


 * I have added a citation of an interesting survey article by Hildebrandt, available online:
 * What a menagerie! But interesting reading, and some coverage of Daniell. The Springer online Encyclopaedia of Mathematics also has a helpful article. --KSmrqT 01:48, 20 July 2007 (UTC)
 * What a menagerie! But interesting reading, and some coverage of Daniell. The Springer online Encyclopaedia of Mathematics also has a helpful article. --KSmrqT 01:48, 20 July 2007 (UTC)

Post flux phase
After all significant changes have been made, I think we are going to need a disinterested and non-technical review of our progress (does the article make sense*(mildly intelligible) to a non-mathematician). I have one person in mind to do this, but if someone has a suggestion, I am open to it.--Cronholm144 02:03, 20 July 2007 (UTC)

tagged partition
In the definition of integration à la Riemann, is there a reason for defining it with partitions of the interval [a,b] in subintervals of different width? Jakob.scholbach 00:38, 23 July 2007 (UTC)


 * (One reason.) For continuous functions on a compact set, it doesn't matter which definition you use.  But it isn't too hard to cook up problematic discontinuities which make a function integrable in one sense but not the other.  For example, the characteristic function of the rationals on [0,1] is not Riemann integrable.  If uniform partitions are used, its integral is 1, which is clearly not what we should expect it to be in the intuitive sense that the function is "almost always" zero.  Silly rabbit 01:05, 23 July 2007 (UTC)


 * Oh, it's embarrassing. Thanks. Jakob.scholbach 01:31, 23 July 2007 (UTC)

Applications???
The article currently does not have a single word devoted to applications! (the section "Methods and applications" only deals with the former). I'm not really into these things, but we should say at least something about: Jakob.scholbach 01:28, 23 July 2007 (UTC)
 * differential equations
 * integral equations
 * what else?