Talk:Improper integral

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I think this page is reasonably 'mature' now in its content; but that the topics could usefully be re-ordered.

Charles Matthews 09:07, 5 Nov 2003 (UTC)

There is now a free-standing Cauchy principal value page: example here -> there?

Charles Matthews 10:45, 14 May 2004 (UTC)

The term "improper integral" seems more widely known that "Cauchy principal value" or "principal value", so if the pages get merged, I think it would be better to keep this one and make the other one a redirect page. Michael Hardy 01:02, 20 Sep 2004 (UTC)

[edit] Introduction

The introduction on this article needs a lot of work. I can hardly understand it as it currently is, so somebody unfamiliar with the material could be hopelessly lost.

  1. The definition of an improper integral is vague and as stated and is either not well defined or a super class of all definite Riemann integrals. (depending on how you read specified real number)
  2. There are multiple topics addressed with little or no transition making reading it misleading (as just before the third integral)
  1. There's too many specific details that should be in the body of the article, not in the intro (as three fourths of it talks of Lebesgue integration) GromXXVII 19:51, 7 November 2007 (UTC)

You added a "definition" that referred to "with integrals of functions that could possibly have infinite area", and you complain of vagueness? How vague can you get?

Can you be specific about what it is that you can't understand about the definition? Michael Hardy 21:26, 7 November 2007 (UTC)

In calculus, an improper integral is the limit of a definite integral, as an endpoint of the interval of integration approaches either a specified real number or ∞ or −∞ or, in some cases, as both endpoints approach limits.
  • Lack of explanation as to the concept, it jumps right into the explicit limit definition (without explicit definitions)
  • approaches either a specified real number could mean any real number, which would imply all definite integrals are improper integrals, or some unknown specified real number the article never tells the reader about.
Essentially, the statement doesn’t tell the reader what an improper integral is, or why we need it. Then it leads into a bunch of cases without an explicit definition.
I was trying to link the two types (unbounded function, unbounded interval) into one cohesive, concise expression. Then it can be further refined into the different types and explicit definitions using limits. Sure it wasn’t the best possible, but with the other changes it made the article more readable and explanatory than it currently is and can always be improved upon. GromXXVII 21:41, 7 November 2007 (UTC)

...OK, I've looked over the article from top to bottom. I've expanded the first sentence a bit to include some expressions in mathematical notation for anyone who finds that easier to follow than the words preceeding it. But you really do need to be specific about each of your complaints. (1) What specifically do you find hard to understand about the initial sentence? (2) In which cases is the concept "not well defined"? (3) Are you saying it is a "super class of all definite Riemann integrals", and you object to that, or that it is not a "super class of all definite Riemann integrals", and you object to that? (4) What sort of "transition" do you have in mind "just before the third integral"? The point of that sentence is that even in cases where an integral can be defined without taking a limit, nonetheless it can't quickly be computed without taking a limit, i.e., in one sense it may not need to be considered improper, but for purposes of computation one treats it as improper. It seems to me that that needs to be given prominence early in the article. (5) Which specific details are you saying should not be in the introduction. Michael Hardy 21:46, 7 November 2007 (UTC)

1) “specified real number” is ambiguous because there are restrictions on what the "number" can be to yield an improper integral. For instance, it implies that \int_0^1 1\,dx = \lim_{t\to 1} \int_0^t 1\,dx is an improper integral. It’s not standard to use a limit in such case, but is certainly possible to do.
This is if the reader assumes “specified real number” could be anything real.
2) It’s not well defined if the reader assumes there are restrictions on the specified number, but they’re not stated. I didn’t mean well defined in the mathematical sense, I meant it as in “part of the definition is not available to the reader”
3) I’m saying it’s not a super class of all Riemann integrals, but the intro makes it seem so. (Also note it certainly doesn’t need to address this, just to be written so that it doesn’t have that tone)
4) I was referring to this, where the integral appears to be trying to help explain the paragraph, but doesn’t quite have that purpose
“In some cases, the integral from a to c is not even defined, because the integrals of the positive and negative parts of f(x) dx from a to c are both infinite, but nonetheless the limit may exist. Such cases are "properly improper" integrals, i.e. their values cannot be defined except as such limits.
This should be reworded, but also in a different section.
The integral
\int_0^\infty\frac{dx}{1+x^2}
can be interpreted as
\lim_{b\to\infty}\int_0^b\frac{dx}{1+x^2}=\lim_{b\to\infty}\arctan{b}=\frac{\pi}{2},”
5) I think that everything after “(see Figures 1 and 2)” can be moved somewhere else.
Ultimately I think it’s most important for the introduction to give a clear idea of what an improper integral is (which would address that a limit or other advanced techniques are used because standard Riemann integration does not work), and when an expression is an improper integral (which would address the issues of the interval the function is integrated over, and the issues of the range of the function)
There are perhaps other things that merit prominence to be in the introduction, but most of what is there doesn’t, especially because there isn’t a clear description currently, so all the stuff about computation, Lesbegue integration, and properly improper integrals serve to confuse instead of inform - even though they should be in the article.
It seems necessary to me that the article quickly:
  • Motivate why improper integrals are used
  • Explain when they’re used
  • Define the symbols used for them (in at least the two simplest cases of an unbounded function and interval). I had four explicit definitions to cover the most common cases so I put them all in a different section.
I think that the version I had would be a better place to continue: with fixing the intro and figuring out what else should have a prominent role. (because as is the introduction seems to have no structure, coherence, and doesn’t give the reader a good idea of what an improper integral is; and most of what is there doesn’t help with that)
I’ll go ahead and work on that unless if you’re just going to revert everything again… —Preceding unsigned comment added by GromXXVII (talk • contribs) 23:00, 7 November 2007 (UTC)
I've created more sections. A standard way to work on articles is to take one section at a time and deal with its issues. Charles Matthews 07:15, 8 November 2007 (UTC)

[edit] A suggestion

The article is currently written so as to accomodate both the Lebesgue and Riemann integrals. This has the effect that it would be, I think, completely impenetrable to someone who is not familiar with the Lebesgue integral. Since the article begins "in calculus", I think this is undesirable.

At the same time, the article is arguably not general enough. Whoever wrote that the improper integral is "not a kind of integral" was not familiar with the Kurzweil-Henstock, or generalized Riemann integral. Among the virtues of this integral is that it integrates all Lebesgue integrable functions and improperly Riemann integrable functions "automatically". Coming back down to earth, it seems clear from linguistic use, interpretation in terms of area and so forth that the improper (Riemann) integral is "some kind" of integral, right??

I recommend that the body of the article be written explictly in terms of the Riemann integral. Here a key fact is that any Riemann integrable function on [a,b] is bounded, which explains why a new definition is needed when the function blows up at one of the endpoints. It seems more clear to discuss separately the case of an infinite interval and an unbounded function. More examples would also be nice.

There can then be a section discussing the difference between the improper Riemann integral, and the Lebesgue integral, and ideally a final section discussing the Kurzweil-Henstock integral.

Again, remember that this is an article that second semester calculus students will try to read. Even the brighest calculus students do not know measure theory... Plclark (talk) 11:35, 20 November 2007 (UTC)Plclark

[edit] sigh....

OK, I'm going to return to this article soon. Right now we find this sentence:

The Riemann integral as commonly defined in calculus texts is only defined for a continuous function f on a closed and bounded interval [a,b],

That is not true. What is true is that introductory calculus texts often consider only the case where the function is continuous. But the definition they state works for many discontinuous functions as well, including piecewise continuous functions of a sort often dealt with in calculus texts. I think it's appropriate to consider in the article the manner in which the concept is first introduced in calculus texts, but this continuity issue is really not the essence of the reason why there is such a concept as that of improper integrals. That's one of a number of reasons I've flagged this for attention. Michael Hardy (talk) 21:07, 4 June 2008 (UTC)

Actually, I changed the phrasing because I was prompted by your "Attention" flag. You are right that the statement is incorrect: it should say that the Riemann integral is commonly defined in calculus texts for continuous functions. (Not that this definition only applies to continuous functions, as is suggested by the current wording.) Unfortunately, there is no elementary characterization of Riemann integrable functions suitable for an introductory section. A function is Riemann integrable iff it is bounded and continuous almost everywhere, but this is quite a bit to put in the first sentence of the motivation. siℓℓy rabbit (talk) 00:36, 5 June 2008 (UTC)

Maybe it would be better to avoid any discussion of different definitions of integral (e.g. Riemann versus Lebesgue) in the early parts of the article. There seem to be two essential issues: (1) some integrals can't be defined except as limits as a bound approaches some limit, and (2) some integrals, even if they can be defined, cannot conveniently be computed except as limits as a bound approaches some limit. In either case, an improper integral is a limit as one or more of the bounds of integration approaches some limit. Michael Hardy (talk) 00:54, 5 June 2008 (UTC)

Yes, that sounds reasonable. In fact, it is probably easier to simply define an improper integral from the outset as the limit of an integral rather than to attempt to motivate it by considering unbounded functions/domains. The motivation is currently more of a potential source of confusion than anything else. The situations in which one would want to (or need to) use an improper integral should still be mentioned, but perhaps after a definition has been properly formulated. siℓℓy rabbit (talk) 03:26, 5 June 2008 (UTC)