Talk:Riemann integral

From Wikipedia, the free encyclopedia

Mathematics Portal This article is within the scope of WikiProject Mathematics, which collaborates on articles related to mathematics.
Mathematics rating:	B Class	Mid Priority	Field: Analysis
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Needs some basics historical information, when was it introduced. Some more comparison with Lebesgue integral would compete a nice article. --Salix alba (talk) 16:46, 18 April 2007 (UTC)

[edit] Animated image

Hello. Riemann integral is a very nice article. I wonder about the animated image. I agree that it's useful for viewing in a browser, but I don't think it would be as useful on a printed page. I don't see a good reason to rule out printed copies; this article could easily be something a student would print out for reference, for example. Can we convey the same information by displaying each frame of the animation separately? -- Sequences of graphs are going to be pretty common in the math articles, and so I think it's important to set a good precedent here. I look forward to your comments. Wile E. Heresiarch 16:38, 21 Jun 2004 (UTC)

Personally I think the animated image is great. It really gets accross the idea well and takes advantage of what the web format can offer. I do see your point that in print it would not work. So the answer would not be to remove the animation from the web version, but to make sure it translated to a series of images perhaps in a future print version. - Taxman 20:38, Jun 21, 2004 (UTC)

[edit] Older discussion

Maybe needs paragraph about Stieltjes generalization. Or perhaps link to separate article about same.

Done

Do you mean totally bounded functions and all that stuff? Loisel 07:30 30 Jun 2003 (UTC)

[edit] Image

Isn't it far TOO BIG?

[edit] 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. —Preceding unsigned comment added by Igny (talk • contribs)

I would not be opposed to it. But I think you would need to also write your proposal on talk:Riemann sum and maybe even contact User:Icedemon who had put a lot of effort into that one. Oleg Alexandrov (talk) 18:19, 5 December 2005 (UTC)

I was not aware that a Riemann integral site existed. I agree that hte two should be merged, perhaps entering the work I did on Right/Left/Middle/Trapezoidal into the Examples section of the Riemann Integral page. Icedemon 03:29, 7 December 2005 (UTC)

Well, in this case I would not be for the move. The article Riemann integral is rather well-written and self-contained, while the article Riemann sum has a lot of wording which I find reduntant. All those sections on left Riemann sum, right Riemann sum, middle Riemann sum could be described just in a couple of sentences instead of several screenfuls of text. In fact, I already did that, see the ==Defintion== section in Riemann sum. So I would agree to move from Riemann sum to Riemann integral just the picture and a paragraph or two; most of the work there is reduntant. Otherwise, I would suggest leaving thing the way they are. Oleg Alexandrov (talk) 04:26, 7 December 2005 (UTC)

I agree that Left, Right, Middle, and Trapezoidal sums can be explained in a few sentences, but not to anybody that is not a mathematician. Please remember that the goal of Wikipedia is not only to be succint, but also to explain the subject in a manner that is understandable by everybody. I have attempted to do so in Riemann sum, and you still refer to my work as "redundant". Redundancy is necessary, in this case, in order to properly explain the material to anybody that does not have a degree in mathematics. As for hte merge, I maintain that either the Riemann Sum Left,Middle,Right, and Trapezoidal be moved in their entirety to Riemann integral or the pages remain separate. Icedemon 00:32, 9 December 2005 (UTC)

That's what I am saing too. The material at Riemann sum is way too detailed to be included in Riemann integral. So then the two articles should be separate. Oleg Alexandrov (talk) 01:29, 9 December 2005 (UTC)

[edit] LaTeX formatting

Who tagged this page as needing LaTeX formatting, and what do you need done? –Ryan McDaniel 15:02, 15 March 2006 (UTC)

You can tell who put the tag from the article history. The work to be done is converting html formulas to TeX formulas, which I think is not urgent at all, or maybe not even necessary. I'd suggest the template be removed from this talk page. Oleg Alexandrov (talk) 02:29, 16 March 2006 (UTC)

[edit] direct proof of f *g is Riemann integrable?

Given that both f and g are real functions, Riemann integrable on [a,b], then how to prove DIRECTLY that f * g (f times g) is also Riemann - integrable on [a,b]? —The preceding unsigned comment was added by 142.104.2.101 (talk) 18:50, 16 March 2007 (UTC).

[edit] Original definition

This article is really well-written and interesting. Does this article give the original definition of Riemann?

In most textbooks, e.g. Cohn: Measure theory, the equivalent definition by Darboux is given and they still call it the Riemann integral.

The Darboux definition is easier for a student to understand. Perhaps the main article should give the easier definition by Darboux, and then another article should give the original definition by Riemann. Pierreback 00:56, 25 March 2007 (UTC)

[edit] Link to Java program

I included the following link:

[[1]] Interactive computer program about the Darboux integral (in German)

I think it is a really nice java program which gives a feeling about the definition. Unfortunately it is in German and has now been removed (by another user) from the article. In my opinion it is good enough to be included. What is the general opinion? 83.253.10.236 23:55, 25 March 2007 (UTC)

[edit] Merger proposal

Rectangle method and Riemann sum should be merged into Riemann integral. Rectangle method is about the method of finding a Riemann sum; it is not a long article and could be its own section in Riemann integral. Riemann sum is the same thing as Riemann integral, with a simpler name. (See above for earlier discussion). Mazin07^C₪_T 15:49, 7 April 2007 (UTC)

I don't think that all those facts about left Riemann sum, right sum, etc, and all those pictures from Riemann sum belong at Riemann integral. That's too much detail. In Riemann integration all that matters is that the Riemann sum converges regardless of how the points inside the intervals are chosen.

So if a merger is done, I think very little from Riemann sum should make it in Riemann integral.

My preferred solution would be to keep things the way they are, and refer to the Riemann sum article in the "Riemann sum" section on Riemann integral, for people who want extended knowledge of the sums. Oleg Alexandrov (talk) 16:13, 7 April 2007 (UTC)

I'd agree that Riemann sum goes into too much detail for this article, and there is even more to say about Riemann sums if you want to. It does make sense to merge Rectangle method and Riemann sum, but I'd keep them separate from Riemann integral. -- Jitse Niesen (talk) 04:16, 11 April 2007 (UTC)

Rectangle method would be much better merged with midpoint method than Riemann integral. 69.231.241.75 06:12, 29 June 2007 (UTC)

[edit] Remember the Reader

I looked at the article on Riemman sum, and it's done up in advanced formal notation. While there are, of course similarities, Rectangle and Riemman are in in completely different leagues; not to mention the significant historical and pedagogical differences. To give an other example, playdoh and Riemman's contributions to differential geometry are related, and deformable solids such as playdoh (or some unspecified dough) are often used to convey the idea of smooth continuous deformation; however, it really just wouldn't be appropriate to lump (no pun intended) playdoh and non-euclidian geometry together in the same article; instead you have links pointing to each. And likewise, the Riemman integral could point to and be pointed to by the Rectangle article.

A merger could hold back Riemman (properly in the realm of advanced mathematics), and would certainly add luggage to the Rectangle article (a famous person, familiar with non-euclidean geometry once said: 'as simple as possible, just not too simple' [paraphrased from the original german])--ElectronicsEnthusiast 00:28, 22 July 2007 (UTC)

[edit] Unencylopedic tone and scope

This piece is very nicely written, but it has a didactic quality which is not appropriate for an encylcopedia article. It reads like a set of lecture notes, often saying "we see" and giving first provisional definitions and then after some discussion fixing them up to give the correct definition.

The article is also sorely lacking in historical information. References to Riemann's original papers are required. As another example, the claim that the Riemann integral is the first rigorous definition of an integral needs to be supported and explained by a comparison to the concept of integral used in the 200 years between the founding of calculus and the Riemann integral. Plclark 04:07, 30 September 2007 (UTC)Plclark

If you don't like it, be bold and fix it: Write out all the "we see"s and find references to Riemann's papers. I agree that the article sounds didactic, but when I wrote it (I'm responsible for most of the text) I figured this was the most comprehensible way to describe the Riemann integral to non-specialists.

I don't think the claim that the Riemann integral is the first rigorous definition is correct, though I did hear it made somewhere; what is more likely true is that it is the first definition of an integral that is independent of the notion of the derivative. My understanding is that before Riemann, the Fundamental Theorem of Calculus was taken as the definition of a definite integral. However, I'm no historian and I could be wrong (even completely wrong) about this. 141.211.62.20 00:43, 4 October 2007 (UTC)

[edit] Continuity

I want to point out that the discussion on continuity requirements for Riemann integration refer to measure theory. This seems unfortunate as one reason for using Riemann's approach is to avoid measure theory! Is there a concise and precise way to describe the requirements that does not use measure theory? Acolombi 17:07, 15 October 2007 (UTC)

Yes and no. The condition, precisely stated, is that the set of discontinuities has measure zero. But the concept of measure zero (with respect to Lebesgue measure) can be defined independently of the whole framework of measure theory: A measure zero set is simply one which can be covered by a set of balls B(x_i, ε_i) such that the sum of all ε_i (2ε_i if you're pedantic) can be made arbitrarily small. Perhaps that should be added to the article on almost everywhere. 141.211.62.20 01:11, 18 October 2007 (UTC)