Talk:Integral/Archive 3

message: 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:
* Discussion/mention of differing definitions of integral (Riemann/Lebesgue, etc.)
* Mention of fields of application.
* Mention of techniques for computing integrals.
As always, any help in adding that succinctly and elegantly to the lede is most welcome. -- Leland McInnes 16:14, 6 June 2007 (UTC)
replies: text: Notice that "lede" is not a word. The word is "lead" as in Wikipedia:Lead section. JRSpriggs 10:01, 10 June 2007 (UTC)
replies: text: 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)
thread_title:  Cleaning up the lead 

message: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Informal discussion 

message: 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)
thread_title:  Introduction 

message: 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, & Nash is one; Stoer & Bulirsch is another. Maybe also the new Dahlquist  & Björck, which is currently available online. Suggestions welcome. Enjoy. --KSmrqT 23:21, 14 June 2007 (UTC)
replies: text: 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)
replies: text:  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)
thread_title:  Numerical quadrature 

message: 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)
replies: text: 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)

text: A very good suggestion. Feel free to implement it. Loom91 08:39, 18 June 2007 (UTC)
replies: text:  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)
thread_title:  Improper integral 

message: 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)
thread_title:  Darboux integral 

message: 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)
replies: text: 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)
thread_title:  Removed a Harvard reference to Rudin's book 

message: Take a look at Henri Lebesgue#Lebesgue's theory of integration. 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)
replies: text: 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)
thread_title:  History of integration 

message: 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)
replies: text: 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 (τ • γ) 12:04, 28 June 2007 (UTC)

text: \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)
thread_title:  Example of nasty integral 

message: 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)
replies: text: 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)

text: 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 (τ • γ) 12:03, 28 June 2007 (UTC)
thread_title:  Notation in improper integral limits 

message: 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)
replies: text: 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)

text: 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)
replies: text:  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)
thread_title:  New lead 

message: Should we include a short introduction to this which leads to the Lp space article? Xantharius 16:53, 11 July 2007 (UTC)
replies: text: 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)
thread_title: L<sup>''p''</sup> spaces

message: . . . 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 · d r is a scalar, since F · d r is the dot product of F with d 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 · d 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)
replies: text: 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)
replies: text: 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)
thread_title: The range of the functions and the range of the integral match

message: 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)
thread_title:  Post flux phase 

message: 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)
replies: text: (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)
replies: text: Oh, it's embarrassing. Thanks. Jakob.scholbach 01:31, 23 July 2007 (UTC)
thread_title:  tagged partition 

message: 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:
* differential equations
* integral equations
* what else?
Jakob.scholbach 01:28, 23 July 2007 (UTC)
thread_title:  Applications???