Talk:Integral/Archive 4

History
The history section needs a careful vetting. It retains the basic outlines given to it by the creator of the section, and appears to give a fairly subjective description of the history of integration, to the point of OR. For example, while there is no doubt that ancient Egyptians asked, and sometimes answered, questions about areas and volumes, is it commonly considered to be "integration"? Likewise, it's better not to go into priority questions (Gregory vs Newton vs Leibniz), and refer to other articles for the fine details of FTC, invention of calculus, and so on. Since this is an article about integral, and not these other subjects, we can afford doing so! Arcfrk 23:22, 8 August 2007 (UTC)


 * I wrote the first deaft of the history section. The Egyptian stuff can potentially stay; it's only a sentence that was cribbed from History of Calculus. Certainly the material on the method of exhaustion should be there. Wrt the stuff about Gregory -- I'm not sure when that slipped in there, but its rather undue weight to the topic. I believe Barrow is also cited as having had some early insight, but again, its undue weight to get into such hair splitting here -- it can go in History of Calculus if it needs to go anywhere. -- Leland McInnes 13:25, 9 August 2007 (UTC)

Just to clarify: by "vetting" I mean checking the text against printed authoritative sources on history of mathematics, not against other wikipedia articles or MacTutor and other compilatory web resources. Besides obvious problems with circularity, the present quality of scholarship even at the better websites is only in a mediocre to fair range. Arcfrk 18:28, 9 August 2007 (UTC)


 * The sentence about Gregory is already in History of Calculus, right in the lead of that article. This is why I copied it in this article. If Gregory and Barrow contributed, I suggest to add a short sentence about that. It is precious epistemologic information. When written properly, history tells people that masters are not gods, and that science does not come from heaven, but always builds from previous knowledge. Popularizing this concept, even is short summaries, is an important mission, rather than hair splitting. And it doesn't require a lot of space. A short sentence is nothing compared to the length of the history section. Paolo.dL 08:57, 10 August 2007 (UTC)


 * You raise interesting points, but here are a couple of remarks.
 * The article History of calculus does not have a lead. The historical section, indeed, mentions Gregory (although I needed to use the "find" function of my browser, it's not by any means a highly visible reference), but then says
 * Of course, important contributions were also made by Barrow, Descartes, de Fermat, Huygens, Wallis and many others.
 * Consequently, singling out Gregory is giving "undue weight" that Leland McInnes mentioned above. This is a somewhat subtler issue than just the length of the lead.
 * You had not read what appears right above your comment before posting it, or else the meaning completely escaped you. Just because a statement appears in another wikipedia article (quarternary source), it needs not be true, accurate, universally accepted, etc. As the article develops, it's inevitable that there would be a lot of unsourced material put in it; however, at some point it becomes necessary to verify these statements with primary and secondary sources. Arcfrk 18:20, 10 August 2007 (UTC)


 * Being picky :-) How comes it was so difficult for you to find the sentence about Gregory? :-) Yes, the article about history does not have a lead but it has a leading section, and in this leading section, the leading paragraph (not only sentence, but a separate paragraph) of the subsection about "Modern calculus" is fully dedicated to Gregory!
 * Secondary source. More importantly, the bibliographic reference about Gregory, which I have included in my edit and you possibly have not seen, was given in the lead of the article about the Fundamental Theorem of Calculus, where again a whole (short) paragraph was dedicated to Gregory (by the way, in that article Newton and Leibniz are not even cited, and I will insert there a citation ASAP). So, I did not ignore your comment about the quality of other articles in Wikipedia. Of course, the authority of the authors of the book (Marlow Anderson, Robin J. Wilson, Victor J. Katz) can be denied, but they happen to be university professors. Also, they provide a rich list of primary references. I think that you should also trust the people who wrote the history of calculus section in Wikipedia. They are as good as you are. But I am sure you agree that if you don't trust them, then you should provide another reference against their claim, show that your reference is more reliable, and delete their comment about Gregor in the other two articles. Coherence in these three articles is important.
 * Undue weight: true! Now, the important question is: didn't my statement about epistemology and popularization touch you? This should lead you to consider that there is even now an "undue weight" given to Newton and Leibniz. So, wouldn't it be nicer to just start from what I did, refine it if needed, change the subheader, and add at the end of the subsection the short sentence about Barrow, Descartes, de Fermat, Huygens, Wallis and many others, rather than totally rejecting my contribution?
 * Less unbalanced subheader. For instance, the subheader "Modern calculus" would be less unbalanced than "Newton and Leibniz" (curent version) and "Gregory, Newton and Leibniz" (my previous suggestion).
 * In the meantime, thanks a lot Arcfrk for your interesting comment. And, for all of you: remember that I appreciate your work and I will always respect your final decision. With kind regards, Paolo.dL 21:16, 10 August 2007 (UTC)


 * I've got a very busy day, but I will take a few minutes to outline some of the many problems with the history section at large, and Paolo's edit in specific.
 * The latter first, concentrating on this difference. To make discussion easier, here is Paolo's version:
 * 
 * Gregory, Newton, and Leibniz
 * The major advance in integration came in the 17th Century. The first published statement and proof of a restricted version of the fundamental theorem of calculus was by James Gregory (1638-1675) . Isaac Newton (1643–1727) and Gottfried Leibniz (1646–1716) independently developed the theorem in its final form. The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Newton and Leibniz developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions within continuous domains. This framework eventually became modern Calculus, whose notation for integrals is drawn directly from the work of Leibniz.
 * Notes
 * for  from   to   is given as   instead of 314.16 (100π).  --Lambiam 09:19, 20 August 2008 (UTC)
 * for  from   to   is given as   instead of 314.16 (100π).  --Lambiam 09:19, 20 August 2008 (UTC)

A bit worried about one sentence
In the introduction, I just made a change:
 * is defined to be the signed area of the region in the xy-plane bounded by the graph of &fnof;, the x-axis, and the vertical lines x = a and x = b.

This doesn't mention how to define the integral (which is beyond the scope of the intro I think), but is it accurate? I'm also a bit worried that it doesn't mention that this area may not exist, but I don't want laden the intro with technical details, so any ideas? Cheers, Ben (talk) 00:56, 16 December 2008 (UTC)


 * I had the same thought. But I think your wording is best.  To be 100% accurate, we should say that the signed area under the graph is defined to be the integral (not the other way around), and the integral is defined by...[whatever].  Nevertheless, the earlier phrasing "...is equal to..." leaves an uncomfortable ambiguity, and the present wording at least captures the general idea of how one tries to define the integral.  Indeed, one defines the integral precisely so that this is true, whenever it makes sense.   siℓℓy rabbit  (  talk  ) 01:08, 16 December 2008 (UTC)

Notation question
I'm concerned about some of the notation in this article. It's in the Introduction section.

$$ \int_0^1 \sqrt x \,dx = \int_0^1 x^{\frac{1}{2}} \,dx = \int_0^1 d \left({\textstyle \frac 2 3} x^{\frac{3}{2}}\right) = {\textstyle \frac 2 3}.$$

I've never seen $$\int_0^1 d \left({\textstyle \frac 2 3} x^{\frac{3}{2}}\right)$$ used in any of my textbooks or by any of my teachers. Are you sure that this is standard notation? —Preceding unsigned comment added by Metroman (talk • contribs) 06:49, 3 March 2009 (UTC)
 * Well, have a look at Riemann-Stieltjes integral. Although it is correct, I don't see the mathematical advantage in using such notation in the given context. -- PS T  09:07, 3 March 2009 (UTC)

Vector-valued integrals
There seems to be some confusion in the third bullet point under Linearity in Properties of integral. There is little point in requiring the space V to be locally compact: over non-discrete valued complete fields that requirement forces the space to be finite dimensional. In addition, the discussion and conditions imposed indicate a possible confusion between strong ("Bochner") and weak ("Pettis") integrals. Depending on how deep one wants to go, it would make sense to discuss:
 * 1) Integral with values in a Banach space (here need the completeness assumption, but crucially also the norm (necessary condition for integrability is that the norm of the function be integrable)). This is the strong integral, and is a rather straightforward generalisation of the real and complex-valued integrals.
 * 2) Integral with values in a Hausdorff locally convex space (was this the source of local compactness, or does that come from the domain of integrable functions - Bourbaki style?). Here the integral is defined for scalarly integrable functions (f for which the function x →  is integrable for each x* in the topological dual of V. This integral is then a (not necessarily continuous) linear form on the topological dual of V, i.e., belongs to V '*, which is precisely the completion of V equipped with the weak topology. Hence the question of whether the weak integral belongs to V or only in its weak completion, which seems to be alluded to in the present text.

However, I'm not inclined to implement the above changes at that particular point in the article, where they do not properly belong. Would be better to be content making the point there that the various integrals are all linear operators on the (vector) spaces of functions where they are defined. Instead, there should be a short summary section on vector-valued integrals, linking to articles on weak and strong integrals. Stca74 (talk) 21:50, 10 March 2009 (UTC)

Cauchy's definition of the integral
I've been reading a little about different definitions of the integral, and a couple of books mention the "Cauchy Integral" which was formulated before the Riemann Integral and is in fact a special case of the latter where the "tag" of each interval in the partition is chosen to be the left endpoint of the interval. I notice that Wikipedia (and seemingly most other online sources from a quick google) doesn't mention it at all, and Riemann Integral even goes as far to say "the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval." when it seems that Riemann merely generalized Cauchy's integral. Cauchy integral is also a redirect to Cauchy's integral theorem.

I feel like it should be included for completeness, at least from a historical perspective if nothing else. I'm no expert by any means, but I might have a go at making an article. It seems fairly odd that it is not referred to on WP at all so I'm not sure whether to just plow ahead and make an article (and change the Integral and Riemann Integral accordingly). It seems it would be fairly straightforward to include since the definitions are so similar to the Riemann.

So basically I'm just wondering if anyone objects to including this integral in WP, or knows anything about it. (also posting this at Talk:Riemann integral) slimeknight (talk) —Preceding undated comment added 02:38, 25 November 2009 (UTC).

Section on Introduction of the Integral
The portion that introduces the idea of the integral, when evaluating, simply goes to the integral to F(1) - F(0), which then evaluates to 2/3. Should it be mentioned that &int;x1/2dx = 2/3*x3/2 ? MathMaven (talk) 16:10, 6 March 2010 (UTC)

Misuse of sources
A request for comments has been filed concerning the conduct of. That's an old and archived RfC, but the point is still valid. Jagged 85 is one of the main contributors to Wikipedia (over 67,000 edits, he's ranked 198 in the number of edits), and practically all of his edits have to do with Islamic science, technology and philosophy. This editor has persistently misused sources here over several years. This editor's contributions are always well provided with citations, but examination of these sources often reveals either a blatant misrepresentation of those sources or a selective interpretation, going beyond any reasonable interpretation of the authors' intent. I searched the page history, and found 2 edits by Jagged 85 in August 2008. Tobby72 (talk) 20:56, 11 June 2010 (UTC)

Needs definition of square-integrable
I followed a link here, but square-integrable is not defined. Nor is it obvious what it means.

According to Wolfram, f(x) is square integrable if the integral of the |f(x)|^2 dx from -infinity to +infinity is finite.

201.229.37.2 (talk) 11:44, 27 August 2010 (UTC)


 * Yes, more precisely, f(x) is square integrable on an interval (or more general set) if the integral of |f(x)|^2 dx on that set is finite. I changed the redirect . The target mentions also that in special cases we have to distinguish which definition of integral we use.--Patrick (talk) 12:54, 27 August 2010 (UTC)

Too difficult to understand and also to calculate some of the integrals
Well, some of the integrals are easy to have solutions but I found a very difficult integral that I cannot solve it even using the assistant.

this:

∫√(1-e&#178;sin&#178;θ)dθ

can anyone help me???@@@Thanks.218.102.106.24 (talk) 14:30, 7 July 2010 (UTC)

Good approach
This article speaks to the layman and gives a simple example early. Only later in the article does it get to more technical issues. This is helpful to the general population of readers. I wish more Wiki authors followed this example when writing about complex math and science issues. —Preceding unsigned comment added by 99.147.240.11 (talk) 20:10, 3 September 2010 (UTC)

Integrals preserving strict inequality
The inequalities section is great, but for the benefit of non-mathematical scientists it may be worth a passing mention whether integrals preserve strict inequalities. That is, if f(x) < g(x) for all x in [a,b], then:

$$\int_a^b f(x) dx < \int_a^b g(x) dx.$$

For a such a subtle change I have actually found this useful in applications, so I think it is worth putting in the article. However not having studied Lebesgue integration formally I'm not 100% sure if it's always true, so I put it up for discussion. 188.220.4.91 (talk) 21:55, 11 March 2011 (UTC)


 * This fact is true. Suppose that f < g. Then 0 < g &minus; f. Hence 0 < &int;(g &minus; f)dx, and it follows that &int;f dx < &int;g dx. Ozob (talk) 15:19, 12 March 2011 (UTC)

Derivations?
The area of a region is increasing by a rate of $$f(x)$$: which means the vertical distance between (x,0) and (x,f(x)). This represents dA/dx=f(x).

Then integrate the area function A(x), which is the reverse of differentiation and we get the area of a function bounded by a curve and the x-axis. Am I right? Garygoh884 (talk) 01:07, 22 May 2011 (UTC)


 * I am not sure what you are saying, but I think you may find calculus of variations helpful. Ozob (talk) 03:10, 22 May 2011 (UTC)

But what is an Integral?
I am sorry, but this article fails as it does not say in simple terms what an integral is from the beginning.

What you need to do is have a very, very simple definition at the beginning and then work up to the technical stuff later. This enables people to understand at the beginning roughly what it is. If they need to know more, it also informs this learning process and is altogether a good thing.

Can someone who does understand the subject do this? BTW contrasting this with differentation does not help as us maths thickos don't know what that is either (which is why we are here in the first place....) — Preceding unsigned comment added by 131.111.27.50 (talk • contribs)


 * There is an informal definition of a definite integral at the end of the first paragraph. It says that the definite integral of a function ƒ(x) between the limits a and b is "the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b". There is also a diagram illusrating this definition. It is difficult to provide a simpler definition which is at the same time reasonably accurate. Which part of this definition do you not understand ? Gandalf61 (talk) 10:47, 15 June 2011 (UTC)


 * Most of it - I know what most of the words mean (although I hadn't come across the phrase "net signed area" before) but I am not sure what to make of their combination. An example from the real world might be an idea; eg the integral of (example) is (what?), can be found (how?) and is measured in (what?), this can be seen on the graph (illustration). A more general definition is (something like yours above).  — Preceding unsigned comment added by 131.111.27.50 (talk) 16:37, 20 June 2011 (UTC)


 * There is quite an extensive worked example, with a diagram, in the Introduction section. Gandalf61 (talk) 18:18, 20 June 2011 (UTC)

(Integral
area under the curve); True? ==

The following text in the introduction to this article got me wondering:

"[...] the definite integral [...] is defined informally to be the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b."

Given a function $$ f: \mathbb{R} \rightarrow \mathbb{R}_+, \ f \in C^{\infty} \ $$, is "the area under the graph" not also the formal definition? As far as I know, the motivation behind both the Riemann and Lebesgue integrals is measuring areas under curves and irregular volumes in a meaningful way. Furthermore, the ways I've seen Lebesgue/Riemann integrals developed and motivated usually emphasizes that definitions are consistent with areas or volumes.

Bottom line: let $$ f: x \mapsto f(x) \ $$ have the properties as above. Is "the area under the graph of $$ f \ $$ between $$a$$ and $$b$$" not a valid, formal definition of $$ \int_a^b f(x) dx \ $$? If not, why not?

Cheers! Trolle3000  [talk]  05:32, 23 June 2011 (UTC)


 * A single definite integral is *always* the area between the graph and the axis wrt integration.

Another way you could think of the definite integral is as a product of two averages: one is the average length of the infinitely many vertical lines in the region and the other is the interval width (infinitely many horizontal lines in a rectangle representing the area of the region).

A hardly known fact is that all integrals are indeed *line* or *path* integrals. As for Lebesgue theory - it is not required in any form or shape.

71.132.128.219 (talk) 21:16, 23 August 2011 (UTC)


 * I can see some problems with using "the area under the graph" as part of a formal definition of integration:
 * We only have a direct (i.e. non-calculus) way of calculating area for simple geometric shapes such as rectangles and triangles. To formally define the concept of the "area" enclosed by a general curve, you would have to approximate the region under the curve by a set of rectangles (or some other simple shape), add up the areas of the rectangles, then see if there is a limit as the width of these rectangles tends to zero. In effect, this is the formal definition of the Riemann integral - so this just introduces "area" as an intermediate concept into the standard definition.
 * Using "the area under the graph" as a formal definition for both Riemann and Lebesgue integrals does not explain why there are functions that are integrable under the Lebesgue definition but not under the Riemann definition - unles you say that "area" means different things in the two definitions, which is somewhat confusing.
 * It is not obvious how "the area under the graph" definition generalises to related concepts such as arc length integrals and contour integrals. Gandalf61 (talk) 08:05, 23 June 2011 (UTC)


 * I realize this simple definition will not hold when talking about integrals from $$ -\infty $$ to $$\infty$$ or when we're dealing with limits of sequences of functions.


 * However, I think we can agree that a simple, closed, non-pathological curve encloses some well-defined property that could be called area, and that a curve homeomorphic to the real line has some well-defined property that could be called arc length? After all, those properties can be measured with either
 * a piece of string and a ruler, or
 * cardboard, scissors and good kitchen scales.


 * As long as we're talking about functions on compact intervals in $$\mathbb{R}^3$$, why shouldn't we be able to assign physical meaning to the integrals? Trolle3000   [talk]  07:10, 23 June 2011 (UTC)


 * That's what the informal bit in the lead is about. It is no way to go around a formal definition of an integral though, there is more than one definition and the area would be defined by the integral so it is a bit of a tautology. For instance in one definition the area can't be defined if all the points on the graph are one except for the rationals where it is zero. In others there's problems with the function going off to infinity or dealing with a path integral round a point in the complex plane. What on earth is the area under a complex number function? Dmcq (talk) 08:55, 23 June 2011 (UTC)


 * What I think the original poster wants to use is the following statement: If S is the region under the curve and &mu; is Lebesgue measure, then $$\mu(S) = \int_a^b f(x)\,dx$$. That's true.  But the usual way of proving it is to prove that both sides are equal to $$\iint_S d\mu$$.  That points out a philosophical difficulty with this approach, namely that you have to go up one dimension.  So if you want to define surface integrals, then you interpret them as volumes of appropriate regions (specifically, regions in the normal bundle to the surface).  If you want to define volume integrals, then you have to measure hypervolumes.  Etc.  One can do this in principle—Lebesgue n-measure is defined for any n, so the definition is not circular—but it would be, I think, messier than the usual treatment.  And one would still want to have a description of which functions are integrable, limit theorems, and so on, and I think that would be more difficult in this framework because the definition of the integral is so indirect. Ozob (talk) 10:36, 23 June 2011 (UTC)


 * I guess the lead could mention that measure is a mathematical generalization of the concept of area and that might make some of the rest more accessible. Dmcq (talk) 17:12, 23 June 2011 (UTC)


 * I think what really bugs me is the word "informally" - it leaves the reader thinking that there is something more to it than area, and there really isn't - apart from all the math, of course! In the article about integrals at Mathworld it says: "An integral is a mathematical object that can be interpreted as an area or a generalization of area."I like that statement. It is precise, and doesn't leave the reader wanting information. So I propose we edit the text in this article to: "Given a function ƒ of a real variable x and an interval [a, b] of the real line, the definite integral [...] can be interpreted as the net signed area of the region in the xy-plane bounded by the graph of ƒ, the x-axis, and the vertical lines x = a and x = b." What do you say?  Trolle3000   [talk]  17:27, 23 June 2011 (UTC)


 * Mathworld tends to be informal anyway. Even so you left one very important thing out from what Mathworld said - 'or generalization of area'. Without that you are left with the informal bit. There is more to it than area. When a mathematician generalizes a teacup can turn into a doughnut. Dmcq (talk) 17:51, 23 June 2011 (UTC)


 * @Dmcq, what you write is one of the things I love about math, but also the reason I think the formal/informal discussion should be left out of the introduction - no need to complicate things further. When dealing with a function $$f \in C^{\infty}, f: [a,b] \in \mathbb{R} \rightarrow \mathbb{R}_+$$ and an integral $$\int_a^b f(x) dx$$, why make it messier than necessary? In this case, how can you interpret the integral if not as the area below the graph? And why not tell that out loud? Trolle3000   [talk]  22:47, 23 June 2011 (UTC)
 * If f represents velocity, then I am much more inclined to interpret its integral as displacement. To my mind, displacement is a more accurate description of an integral than area.  (For example, the interpretation as displacement is true even if f is not non-negative.)  But even that is only valid in the one-dimensional case. Ozob (talk) 23:45, 23 June 2011 (UTC)
 * Now we are talking physics. If "f(t)" represents (non-negative) acceleration and "t" represents time, then the integral as well as the area below the graph represents speed. If "f(s)" represents (non-negative) force and "s" represents displacement, then both the integral the area below the graph represents work. But we are still dealing with an area, only with other units than "length x length" Trolle3000   [talk]  23:57, 23 June 2011 (UTC)
 * No, once you make those interpretations, the integral is no longer an area. The integral is only an area if f is interpreted as height above the x-axis. Ozob (talk) 10:27, 24 June 2011 (UTC)

History
I have removed the paragraph

That same century, the Indian mathematician Aryabhata used a similar method in order to find the volume of a cube.

since it has little in common with what the cited article states:

The formulas for the sums of the squares and cubes were stated even earlier. The one for squares was stated by Archimedes around 250 B.C. in connection with his quadrature of the parabola, while the one for cubes, although it was probably known to the Greeks, was first explicitly written down by Aryabhata in India around 500

Sasha (talk) 22:57, 2 January 2012 (UTC)


 * This article has been edited by a user who is known to have misused sources to unduly promote certain views edits (see WP:Jagged 85 cleanup). I searched the page history, and found 3 edits by Jagged 85. Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent. Tobby72 (talk) 19:22, 19 January 2012 (UTC)

Needs section on AUC, or split AUC to its own page
"Area under the curve" redirects here, but this page does not define an AUC in terms of its use in statistics or give the reader an indication of how they should interpret an AUC when they first encounter one. — Preceding unsigned comment added by 145.117.146.70 (talk) 10:33, 14 October 2010‎ (UTC)

integration
integration of cos x/sin²x — Preceding unsigned comment added by 41.221.159.84 (talk) 15:58, 18 February 2012 (UTC)

Transport function
I'm not totally sure, as the article on the "transport function" is very short, but I'm pretty sure that the "transport function" is NOT a definition of the integral as is stated in this article on integrals. 24.18.97.156 (talk) 01:23, 11 April 2012 (UTC)

Split?
At the moment this article tries to cover too much. It might be in order to split it into an article on single-variable, real-valued integration (which could then talk much more about applications of these basic integrals), and a more general article on integration, its history, and a list of types of integration written in summary style. — This, that, and the other (talk) 09:45, 13 May 2012 (UTC)


 * I think the length and coverage are about right for a top level article like this. There's already summary style happening in each section.  I don't find the emphasis on any one topic to be overwhelming.  Overall, it's a well-balanced article of an appropriate length.  The main deficiency is better citation style.   Sławomir Biały  (talk) 12:06, 13 May 2012 (UTC)

Everywhere continuous but nowhere differentiable functions
Hi!

According to my sacred texts, any continuous function on the closed interval [a,b] is Riemann integrable over that interval. Now there exist functions satisfying that condition - hence integrable - but nowhere differentiable. So, forgive me my ignorance, but I take this to mean that the integrated function (although it can't be expressed in a closed form) is differentiable, once. It seems a bit screwy. Have I misunderstood something? In any case, might it be worth mentioning integration and these functions in the article regarding Riemann integration? All the best 85.220.22.139 (talk) 16:13, 28 July 2013 (UTC)
 * Whoops! My mistake. I was talking about a definite integral which has, of course, a numerical result. I beg yoyr forgiveness, but I still think that nowhere continuous functions deserve a mentions. All the best. 85.220.22.139 (talk) 17:12, 28 July 2013 (UTC)


 * You might appreciate differentiability class. Ozob (talk) 20:17, 28 July 2013 (UTC)

Recent edit
I am writing here about this edit, whose edit summary reads "Layout/formatting changes and formatting/cleanup templates added. Moved history section to the end of the body and moved an oversized image out of the lead. This page really needs a lead rewrite." My inclination is to revert this edit, since I disagree with everything that it did: -- Sławomir Biały (talk) 11:40, 21 October 2013 (UTC)
 * 1) The edit removed the image from the lead, with no real justification except to say that it was oversized.  It is not an oversized image: in fact its dimensions are quite typical for a lead image.
 * 2) The edit added the template lead rewrite with the justification "The current lead section lacks sufficient generality to summarize all forms of integrals (and hence, the article)".  It may well be true that the current lead does not summarize all forms of integration, however it does support the article as currently written.  All aspects of the article are summarized in the lead, roughly in proportion to their prominence in the article.
 * 3) The edit added the template too many photos.  I can't see how this is remotely the case.  Many sections have a single image in support, the chief exceptions being the section on Riemann integration which has two, and the long introduction section which has three (that comfortably fit within that section).  This does not seem at all to be excessive.
 * 4) Moving the history section to the very end of the article seems to run against the purposes of WP:MTAA.  The history is the most accessible portion of the article, so it should be nearer to the top than to the bottom.


 * That's not everything I did in my edit - you reverted changes that addressed conformity problems with WP:IMAGELOCATION and WP:LAYIM - (see formal definitions section where the line is cut). MOS indicates there are too many images when there's overrun into another section.  I put in a temporary fix with the "clear" template, which you then reverted.  The ideal fix would be to use a gallery to group the images neatly, not delete images.
 * The sandwiched text between the left and right images under Riemann integral is (IMO) the worst MOS flaw/appearance issue on this page. (WP:IMAGELOCATION) If you don't like my fix, you need to do something else to address it.
 * The TOC is extending the lead section due to its excessive size and position - the only solution I know of for fixing that is a TOC limit.
 * Neither the first definition, nor the first paragraph of the lead, sufficiently define the integral conceptually or mathematically in a general context (which would describe the integral of a map over an arbitrary space). At minimum, it should mention more general integrals (i.e. types of integrals and spaces over which one can integrate) in this paragraph.  I do not think it's a good idea to use a mathematical definition of the integral in this general context, because that would be too technical for most readers; however, I think it's absolutely necessary (it's also indicated in WP:LEAD and more specifically in WP:GOODDEF) to adequately describe integration in general terms, not in specific cases.  The current lead would be great for an article on Riemann integration on the real line; but, you'll need to explain to me exactly how the first paragraph reflects upon a Lebesgue integral over an arbitrary/general measure space, because I don't see it.  I can't think of an adequately general definition/description off the top of my head, but it should answer the question, "What does the integral of a mapping actually represent in practical terms?"
 * I think you raised a good point about keeping the history section as the first section though.
 * Also, I misread the source code information on my browser when I checked the lead image - I read the default size (420px) instead of the current size (300px) For future reference, a lead image is "oversized" (by policy definition) if it is >300px, per WP:LAYIM. So that was a reasonable thing to revert.
 * How would you prefer to address these remaining issues? Seppi333 (talk) 17:37, 21 October 2013 (UTC)

I missed some minor formatting changes, but the edit was not adequately summarized. (It would be more helpful to roll this out as a sequence of edits, each with an informative edit summary about precisely what was done rather than relying exclusively on a diff to determine what had changed.) I have fixed the text squashing issue and set the TOC limit to 2.

I don't really follow your point about the lead being too specific. The Lebesgue integral also measures the signed area under the graph of a function, so it's not overly specific to context of the Riemann integral. It would be inappropriate to attempt in the first paragraph to emphasize the general case of an abstract measure space since this is treated only briefly in the body of the article itself. Whether this focus is appropriate is ostensibly a problem with the article, not with the lead. Sławomir Biały (talk) 21:29, 21 October 2013 (UTC)
 * I'll put in a gallery and add content on the Lebesgue integral once I've finished taking amphetamine to FA status. For the lead, I really just meant the definition or description should encompass that kind of integral over that form of space in addition to a Riemann integral on R.  Basically, what's stated doesn't describe the mathematical term "integration" in general - so it's incomplete, not wrong.  I'll contact you for your input on addressing this when I'm ready to work on it (assume it isn't fixed before then).
 * Regards, Seppi333 (talk) 17:25, 22 October 2013 (UTC)


 * I'm not sure what content you had in mind with respect to the Lebesgue integral: the lead already mentions that way of integration and, moreover, already includes the basic intuition that is common to both Riemann and Lebesgue integration. For readers wishing to know more about these different notions of integration, there are actually separate articles Lebesgue integral and Riemann integral.  However, there are many other kinds of integrals: for instance the Denjoy integral, the Henstock–Kurzweil integral, the Daniell integral, the Gelfand–Pettis integral, the Riemann–Stieltjes integral, the Lebesgue–Stieljes integral, and so forth.  So it's not at all clear what your ideal lead should look like, nor how you could possibly reference such a lead that encompasses all of these standard generalizations of the usual integral (nor whether such an attempt would actually be an improvement for the typical reader of this article).


 * While I would enjoy immensely a serious attempt to clarify to another mathematician what the noun "integral" actually means, I should caution that there is ostensibly a plethora of different notions of "integral" that might be of interest to, say, a high school student, an undergraduate major in the sciences, a mathematics major, a graduate student, or a mathematics researcher. This article, as a top-level article on the topic, should probably cater to the least common denominator of this group.  The most common intuition is the area under a graph, as the lead already discusses, and this intuition is actually valid for both the Riemann and Lebesgue integrals.  A specialist interested in a particular kind of integral should be able to navigate easily to more specialized articles (there are links in the text as well as navboxes and categories) whereas a novice needs a description of the topic that is familiar and easy to understand.


 * The bottom line is that if you find that our treatment of the Lebesgue integral is lacking, then the appropriate article to edit is Lebesgue integral, not necessarily the main page Integral, just as if you were to feel that we did not adequately address the difference between the Henstock&endash;Kurzweil integral and the Daniell integral, for example, then the appropriate place for that discussion would be on some subordinate article.  Sławomir Biały  (talk) 00:45, 27 October 2013 (UTC)


 * Hi Slawomir, I'm still not ready to work on this yet, so I won't be able to really follow up - but the very least that I can say is that when I do work on it, I'd be using WP:RS as is required - my own definition is moot in relation to the topic. I'm well aware that there are many different types of integrals.  That's precisely why such a definition in the lead should encompass integration theory in general.  Regards, Seppi333 (talk) 01:33, 27 October 2013 (UTC)

Typesetting of the differential operator
In the Terminology and notation section, it says "Some authors use an upright d (that is, dx instead of dx)", when ISO 80000-2-11.16 shows that an upright Roman type is written for the differential. Should the article be changed to reflect this? — Preceding unsigned comment added by 94.9.152.183 (talk) 14:10, 19 July 2015 (UTC)


 * No. This has been discussed many times before.  I have always maintained that an upright d is an error, and others have said that they too prefer an italic d.  Additionally, the principle of WP:RETAIN says that we should not make stylistic changes such as this (except to make an article internally consistent).  Ozob (talk) 17:53, 19 July 2015 (UTC)


 * In general Wikipedia goes by usage rather than by standards. I keep on seeing bits being quoted from the ISO 80000 series and disagreeing with what they say, I wonder if no mathematicians were consulted as it says 'to be used in natural sciences and technology'. You'll sometimes see bits in Wikipedia about how things are represented one way in physics and then another way in mathematics. Dmcq (talk) 20:05, 19 July 2015 (UTC)