Talk:Riemann sum

Merge?
I think the Riemann sum and the Riemann integral have too much in common. I suggest merging information from them into Riemann integral, and make Riemann sum a redirect. Please see discussion at talk:Riemann integral.(Igny 21:51, 5 December 2005 (UTC))

Yes I was looking for this article but found that one instead. Very confusing. If you knew the difference between a Riemann sum and an integral why would you need to look it up?Circuitboardsushi (talk) 21:56, 7 April 2012 (UTC)

This is also confusing because the first figure goes into a lot of detail about maximum and minimum methods. This seems like a bad idea because these concepts are not discussed on this page, but rather appear on the Riemann integral page. OceanEngineerRI (talk) 20:27, 28 August 2018 (UTC)

Clarification
I think it would be worth clarifying that the distance between the points (x1, x2, xi-1, xi) have to be a uniform distance, and it is done for simplicities sake. The graphs would give that idea as well, which isn't really true of Riemann's original "Riemann sums." --AstoVidatu 04:47, 7 December 2006 (UTC)

==Error Estimation==

Are the error estimation formulas correct for the "middle sum" and "trapezoidal sum" methods? The "middle sum" error estimate is currently quoted as :$$\left \vert \int_{a}^{b} f(x) - A_\mathrm{mid} \right \vert \le \frac{M_2(b-a)^3}{(24n^2)},$$

...and the "trapezoidal sum" error estimate is :$$\left \vert \int_{a}^{b} f(x) - A_\mathrm{trap} \right \vert \le \frac{M_2(b-a)^3}{(12n^2)},$$

I don't have a calculus book handy, but it doesn't make intuitive sense that the "trapezoidal sum" error could be twice the size of the "middle sum" error. Is this right? Is there a handy reference online where these formulas are derived?

--Imperpay 22:30, 26 March 2007 (UTC)

The formulae are correct. http://people.hofstra.edu/stefan_Waner/realworld/integral/numint.html Accuracy of Trapezoid and Simpson Approximations lists the formula for the trapezoidal error as $$\left \vert \int_{a}^{b} f(x) - A_\mathrm{trap} \right \vert \le \frac{M_2(b-a)^3}{(12n^2)},$$

It may not be intuitive, but oddly enough, the consideration of multiple derivatives and the fact that the middle sum method overlaps in both directions makes the error bound for the middle sum method smaller.

-- Icedemon —Preceding unsigned comment added by 98.226.21.88 (talk) 09:35, 25 December 2007 (UTC)

Examples
how have you guys not added the limit of the Riemann sum, which solves the integral with no error? can someone please add this vital information? --69.125.25.190 (talk) 23:49, 1 November 2008 (UTC)

I've added an example but I need some help with the formatting. If someone could clean it up for me, that would be great. Thanks. Dwees (talk) 04:53, 26 November 2008 (UTC)

Would you be able to include sigma notation of the series in the written the examples? Stevescott517 (talk) 03:09, 30 July 2015 (UTC)

Subsets or elements?
Regarding: "Because P is a partition with n elements of I, the Riemann sum of f over I with the partition P is defined as" Should that be "n subsets of I" instead of "n elements of I"? 76.175.72.51 (talk) 17:06, 14 October 2009 (UTC)

Simpson's Rule is Blank!
There are examples of using Simpson's Rule, but there is no theory under the blue heading like the other Reimann sum methods, just examples. —Preceding unsigned comment added by 134.114.119.6 (talk) 03:14, 14 March 2010 (UTC)

This article should reference Archimedes' use of Reimann sums
The article located here http://en.wikipedia.org/wiki/Archimedes_Palimpsest states, "When rigorously proving theorems, Archimedes often used what are now called Riemann sums."

I think Archimedes' use of this method should be noted in this article so as to make it clear that Reimann did not originate Reimann Sums. — Preceding unsigned comment added by 50.46.144.101 (talk) 11:34, 24 December 2011 (UTC)

Merge in Rectangle method
The Rectangle method and Riemann sum are the same; I propose that they should be merged. Klbrain (talk) 20:55, 26 April 2016 (UTC)


 * The Riemann sum is a more technical and complicated way of explaining integrals. I do believe, however, that Rectangle method should be merged instead with Trapezoidal rule. Both these quadrature methods are very similar and at the same level. ReallyFat B. 11:46, 17 May 2016 (UTC)


 * Support merge with Riemann sum. Oppose with merge Trapezoidal rule. The "rectangle method" (a name I've never heard used in seriousness) is, as proposer suggests, exactly the same as Riemann sums. The Trapezoidal rule however is the first in a series of methods that attempt to improve on the accuracy of Riemann sums (with Simpson's rule etc following as the order of the function at the "top of the box" gets higher). The trapezoidal rule is literally not a rectangular sum and therefore it should not be merged there. Jason Quinn (talk) 17:38, 11 June 2016 (UTC)


 * I still support the merge of Rectangle method to Riemann sum (in agreement with Jason Quinn). The argument that the Riemann sum is a "more technical and complicated wat of explaining integrals" is, in some ways, an argument for merging Rectangle method into Riemann sum, making the final article statisfy WP:ACCESSIBILITY. In general, pages covering the same topic with different level of complexity are better discussed together. I can't see that merging Rectangle method and Trapezoidal rule would work as they are different methods, unless they were both merged to Numerical integration (which is the relevant target from the Quadrature dab page) (and it was then argued that all such methods should be on one page).Klbrain (talk) 13:03, 27 September 2016 (UTC)

✅ I've gone ahead and performed this merge. I saw no text immediately incorporable so I merge no text. Some of the error stuff might be useful but would need to be customized for this article. Jason Quinn (talk) 21:52, 27 July 2017 (UTC)

trapezoids etc.: completely wrong
That phrase of the introduction,
 * The sum is calculated by dividing the region up into shapes (rectangles, trapezoids, parabolas, or cubics) that together form a region that is similar to the region being measured...

is totally wrong and misleading. The Riemann sum is by definition calculated using rectangles only. The phrase refers to methods for approximating the Riemann integral, not the Riemann sum. Also, "region" is misleading, this word in this context should refer to the region of integration (which can be 1-D, then there are only segments ~ rectangles, noting else) or 2-D (then one could use rectangles or triangles to divide up the domain of integration, but not parabolas etc.) or higher dimensional (see finite elements). However, the "author" appears to call "region" the "area below the graph of f" (to simplify). I hope others agree, and then this should be fixed (maybe simply deleted, or be moved into a section "approximation of Riemann integral" further down). &mdash; MFH:Talk 13:50, 28 February 2019 (UTC)


 * Senior Professor of Mathematics here. The objection above is completely valid and I am shocked and dismayed by the introduction to this article. Usually Wikipedia is pretty good on math topics, but this is a stand-out counterexample! 66.183.205.13 (talk) 18:32, 13 January 2024 (UTC)

Methods
Is it just me or is the minus sign in $$\Delta x = \frac{(b-a)}{n}$$ not appearing in the Methods section? Why is that? The markup looks perfectly fine, and it's working here ... - Astrophobe  (talk) 04:55, 24 December 2020 (UTC)

Trapezoidal rule
I don't know if it is worth noting that the trapezoidal rule gives sums that are Riemann sums. The partition won't be uniform, but it absolutely falls into the class of Riemann sums, as do more complex rules like Simpson's rule. (Even Gaussian quadrature can apparently be written as a Riemann sum.) At the very least we should avoid saying that the trapezoidal rule is not a Riemann sum.

MathHisSci (talk) 09:17, 5 June 2021 (UTC)