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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)
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)
In the lead of this article, the history is summarized with poor wording. According to the "History" section, the concept of integration was not "formulated by" Newton and Leibniz. They formulated the concept as we know it today, using for the first time a "systematic approach". They "generalized" and "formalized" it in a way, but the concept itself, in a less systematic and less general form, was formulated centuries earlier, as written in the History section. I tried to improve, but someone very unpolitely just reverted my edit without explaining nor trying to understand my rationale, which I explained clearly in my edit summary... Paolo.dL 07:46, 10 August 2007 (UTC)
Thank you, your edit improved the sentence. I understand and accept your rationale. Paolo.dL 21:16, 10 August 2007 (UTC)
Forgive me if I'm wrong, but wasn't the fundamental theorem of calculus (which by the way, was better-written and more accurate than the one on that page itself) specifically requested for this article? I spent a while getting a completely correct statement down as generally requested. I understand that the article is long, but having an article on integration without this was deemed incomplete. And now it's no longer there. Thoughts? Xantharius 15:43, 9 August 2007 (UTC)
Now that the article has stabilized a little, perhaps it's time to think about the Introduction. One of the first things that I feel should happen is the elimination of the plural first person and the conversational tone, as per the Wikipedia Mathematical MOS. This can be a hard thing to do, though. Also, I'm not sure about being asked to consider a swimming pool. What if I don't want to? (Half-joking.) Further, this is a good analogy, perhaps, but swimming pools often just have linear slopes, which therefore don't require calculus to find their volumes.
Perhaps we can get some consensus before going to town on this, and then do and evaluate. Xantharius 18:48, 10 August 2007 (UTC)
Why is the introduction restricting line integrals to two to three dimensional functions? Rubybrian ( talk) 14:19, 10 December 2007 (UTC)
Integral is in my view getting quite close to successful WP:GA nomination. The coverage of topics is (at least) sufficient at the moment, and the lead is certainly among the best-developed and accessible among maths articles.
However, I feel there are still certain issues with the overall structure and the flow of the article, issues that have to do with the organisation of mathematical content. I also feel some of these issues would be easier to resolve if the more specialised articles referenced from / referencing to Integral were in better shape and formed a more coherent whole. I have added a preliminary proposal for work on these relating articles to better enable Integral to stay focussed.
I have collected below some obervations and proposals specific to the article Integral itself.
The article mixes different types of integral in a way which is likely confusing to a beginner. There are in a way two broad topics within the scope of the article:
As it stands, the article introduces (without much discussion) the general measure-theoretic integral (calling it Lebesgue integral) quite early. However, after that, the unifying nature (among the topics of type 1 above) of the general integral is largely lost, and various different integrals (Riemann-Stieltjes, Lebesgue-Stieltjes, Daniell) are introduced/mentioned without making the connections clear. One thing I'm afraid can easily happen is that the (general) Lebesgue integral gets confused with integration with respect to the Lebesgue measure. In similar vein, multiple integral (Fubini's theorem, really) is introduced as a way to move beyond integrands defined on more general domains than intervals in R, despite the fact that the genereal integral has already been defined. The "Properties of integral" section discusses linearity three times in increasing generality, then discusses various inequalities that hold for the general integral in terms of the Riemann integral. Finally, the "Extensions" section introduces three quite different concepts on the same level: improper integrals, multiple integrals and (the various types of) integrals of differential forms. Multiple integrals should probably appear earlier, while the integrals of differential forms should really be its own top-level section (preferably a short one that points to another article).
"2 Terminology and notation" and "5.3 Conventions" should probably be integrated with other sections, and Conventions made much shorter.
The History section should cite a few sources. Good ones (also for some expansion) are the Historical Notes in Bourbaki's Functions of a Real Variable (pp. 129-162 of the English edition) and Integration (pp. V.123-136). History should probably be moved later in the article.
Most of what I would suggest doing to the article has to do with restructuring and moving text around, condencing where possible (hopefully leveraging related articles brought in line with this one). Stca74 13:09, 9 September 2007 (UTC)
Question?? 1)why we are using integration??can you give real time examples?? 2)where does it come to the picture?? —Preceding unsigned comment added by 60.254.13.8 ( talk) 18:21, 19 September 2007 (UTC)
Skand swarup ( talk) 13:44, 8 May 2008 (UTC) Integration is used to find areas of figures which are not geometric. Suppose you spill water on the floor and want to find out what area the water has covered, you can do so by integrtion. What it does is that it breaks up the non-geometric shape into a number of tiny geometric shapes. It then calculates the area of each of the tiny figures and adds them up. This of course gives only an approximation to the actual area.
Leland McInnes reverted my attempt to make the article more accessible with the comment
With respect I believe you need to offer more of an explanation. The text I appended to the intro paragraph was the following.
To be frank the article as it stands is a bit "gear-headed". That is, mostly the only people who would understand it would be those who already have a background in calculus. In principle since an encyclopedia is supposed to be generally accessible (Wikipedia even more than most) it is worth at least trying to make the introduction clarify the subject matter for as wide an audience as possible (I'd argue that the goal should be broader than that but at least that is a starting point).
-- Mcorazao 21:52, 12 October 2007 (UTC)
Please comment or I will simply put back my edits. -- Mcorazao 15:15, 16 October 2007 (UTC)
I was making links for my page direct integration of a beam and when I was putting in the link for this page, I noticed that the lead for the article reinforces the perception that an integral only represents "the area under a curve." Since there's not another page on mathematical integration, this is more of an issue in this article. I would think the lead should point out to readers that what f(x)dx represents is a box with width dx and height f(x), and that the integral of it is a summation of these rectangles' areas as dx->0 while x goes from a to b. JW 05:44, 13 October 2007 (UTC)
seeing the wide variety of meanings at integral (disambiguation), and the status of the word integral as a generic English word, I wonder whether it wouldn't be advisable to move this article to integral (calculus) on grounds of the "principle of least surprise". dab (�) 18:21, 13 November 2007 (UTC)
A tag has been placed on the article. I am tempted to just remove it whilst citing Wikipedia:Scientific citation guidelines, but I think that a discussion should take place here first for the sake of completeness. Thoughts? — Cronholm 144 05:39, 13 December 2007 (UTC)
It seems to me that the animation associated with the line integral section is incorrect or at least misleading (I feel incorrect). It does give a good graphical explanation of a mathematical operation; however, the operation shown is not a line integral. As I understand it, a line integral on a vector field returns a scalar value; the 'sum' of the dot products between the unit tangent vector to the curve and the field value at each point. However, the graphic shown just gives the 'sum' of the field values (not dot products) (and is vector valued).
I am only a second year degree student, so it is quite possible there are other definitions of line integration I don't know of, or that I am just incorrect. However, being a degree student I have been directly affected by the animation; not knowing the definition of a line integral, I used the animation as a definition in a set problem (which caused problems for me!). I can say from experience therefore that the animation was misleading and definately unhelpful. If it is a (rather than the) correct definition, a note to this effect would be very useful.
If I don't find a reply or changes to the article, I will remove the image to avoid other people getting the same problems from it I did. I'm also posting this message in the main Line Integral article, which uses the same image. —Preceding unsigned comment added by 88.106.245.46 ( talk) 12:06, 25 December 2007 (UTC)
I don't think that this particular animation adds anything of value. On the other hand, yes, it is very confusing, and should better be removed. Having said that, I am truly amazed that anyone would try to infer a basic mathematical definition from an illustration in the middle of a wikipedia article! Arcfrk ( talk) 17:59, 29 December 2007 (UTC)
This article reads too much like a text book for trained mathematicians. It is largely inaccessible to lay-people in search of general and simplified knowledge of the subject. I found it disappointingly unhelpful. 143.97.2.35 ( talk) 16:07, 27 December 2007 (UTC)
What were you hoping for? Why do you want to know about integration? The article does need some mathematical knowledge to understand, but there is a limit to how simple it can be reasonably made. Do you need to know what integration in general is, how to do it, or about a specific type of integration (line integration for example)? If you are more specific I will try to improve the article.
The juxtaposition of two symbols have different meanings.
This is hard to newcomers. I suggest that the multiplication sign be written explicitely in order to reduce the confusion.
Write
rather than
Any objections? Bo Jacoby ( talk) 22:55, 9 January 2008 (UTC)
Dear Bo,
I'm not sure that I like the idea of because I think it might mislead a reader into thinking that is a number. It is useful to emphasize that is not a product of and , which is why one usually puts a \, between the and , like so:
Loisel ( talk) 03:34, 10 January 2008 (UTC)
I think it is a poor idea, since interpreting f(x)dx as the product of f(x) and dx isn't necessarily correct and might lead one to believe dx actually represented an infintesimal, which is certainly quite wrong for many (or indeed almost all) defintions of integral. Ultimately for many definitions the dx is purely formal notation and not really representative of anything. On those grounds I suggest that "making the multiplication explicit" is actually more misleading than helpful. -- Leland McInnes ( talk) 17:07, 10 January 2008 (UTC)
Well, ladies and/or gentlemen, notational convenience is the same thing as algebra, and there is nothing wrong in that. I was surprised that the disagreement was about the multiplication itself, rather than about the multiplication sign. The simpler introduction to differentiation and integration is to begin with polynomials. If x is a formal variable, then so is dx. The rules of algebra are d(x+y)=dx+dy and d(x·y)= x·dy+dx·y. These rules are sufficient for deriving the rules for differentiating a formal power series and for solving differential equations. The rules of interpretation is that if dx is not zero, then x is neither constant, nor maximum, nor minimum. The interpretation in terms of limits, and the Riemann and Lebesque integrals, are not needed for quite a while. The beginner needs a break. Wikipedia should explain, rather than just repeat unintelligibly advanced stuff. Bo Jacoby ( talk) 15:13, 20 January 2008 (UTC).
The omission of an explicite multiplication sign is widespread. That does not mean that the multiplication sign is a neologism. The article on polynomial omits the multiplication sign, but that does not mean that multiplication is not intended, for example 2xy2 means 2·x·y2. Don't you all agree on that? Omitting the multiplication sign makes no harm until juxtaposition means something else than multiplication. Then confusion appears. For example (f+g)(x) = f(x)+g(x) defines the sum of two functions, f and g. Here the juxtaposition (f+g)(x) does not mean the multiplication (f+g)·(x). Mathematicians don't mind very much, because the parenthesis around (x) indicate that x is argument to a function, but parentheses have other meanings. In the expression (f+g)(x+y) it is less clear whether the interpretation f(x+y)+g(x+y) or (f+g)·(x+y) is intended. Have for example a look on the articles catenary and gamma function and identify which juxtapositions in the formulas indicate multiplications and which ones do not, and why. Explicite multiplication signs sure would help a lot. So, omission of multiplication signs in formulas may be polite to the author, but it is rude to the reader. Regarding integration there is no doubt that Leibnitz intended a multiplication between the function value f(x) and the differential dx: The differential of the area bounded by the x-axis, the y-axis, the curve y=f(x), and the vertical line at x, is the height f(x) times the base dx. The difficulties in interpreting the differentials have historically lead to tricky definitions for derivation and integration, but the algebraic axiomatic approach avoids these complications. You do not need to know what a differential is, as long as you can use it correctly in computations, just as you do not need to know what −3 means, except that is solves the equation x+3=0. Bo Jacoby ( talk) 02:13, 22 January 2008 (UTC).
Dear Lenand McInnes. Yes, we are talking formal infinitesimals. No, I am not defining my own integral. If
does not mean
then you cannot deduce that
because then you cannot rely on the distributive rule of multiplication:
In the expression for the Riemann sum
multiplication is obviously implied
Also the article Darboux integral defines the upper Darboux sum of ƒ with respect to P:
Here too juxtaposition means multiplication:
Mainstream litterature on Riemann integral, Lebesgue integral and Henstock-Kurzweil integral, as well as the corresponding wikipedia articles, generalizes the elementary Leibnitz integral assuming that the readers are already familiar with elementary high-school algebraic integration. This assumption cannot be made here. I thought that the confusion about the interpretation of juxtaposition was confined to beginners, but now I realize that I was wrong. Bo Jacoby ( talk) 11:32, 22 January 2008 (UTC).
I'm just curious. What does the juxtaposition of f(x) and dx in "f(x)dx" mean if it does not mean multiplication? Is the formula d(2x)=2dx not involving two multiplications? Bo Jacoby ( talk) 15:47, 22 January 2008 (UTC).
Thank you for answering. I understand that you consider "d(2x) = 2dx" illegitimate while "d(2x)/dt = 2dx/dt" is legitimate, meaning "d(2·x)/dt=2·dx/dt", involving two multiplications. Am I right? "∫ dx = ∫ 1dx". Right? "∫ f(x)dx = ∫ (f(x)·1)dx = ∫ f(x)·1dx = ∫ f(x)·dx". Right? Algebraic shortcuts among friends seems to be taboo i WP even if they work. The Riemann integral was supposed to generalize the antiderivative, rather than to restrict the algebraic freedom. Bo Jacoby ( talk) 13:56, 23 January 2008 (UTC).
The argument used in Fundamental_theorem_of_calculus#Intuition for writing meaning does perhaps involve handwaving, but the differential algebra does not. There is no handwaving about the rules of differentiation. Differentiation d is defined on the polynomial ring Q[x] by introducing another variable, dx, and the rules that d(x)=dx, and that dk=0 when k is a number, and d(X+Y)=dX+dY, and d(X·Y)=dX·Y+X·dY when X and Y are polynomials. This is sufficient in order to specify formal differentiation on power series, and all the entries in lists of integrals follow. Limits are not needed for the algebra, only for the interpretation. This means that d(2·x)=2·dx is an algebraic fact. It does not rely on Riemann integration. Bo Jacoby ( talk) 16:54, 23 January 2008 (UTC). PS. The lists of integrals says: Are you saying that this elementary formula doesn't represent correct mathematics? I politely request you to tell me which ones of my statements above, that end with "right?" or "am I right?", that you consider wrong. The summary statistics, that most of them are wrong, is not sufficient. I am trying to figure out what you guys mean. Using your (nonstandard) notation Ix(f(x)), I note the rule that Iax(f(x))=Ix(af(x)), showing that a constant factor can be moved between two arguments, which implies multiplication. Bo Jacoby ( talk) 11:52, 24 January 2008 (UTC).
Letting new readers of Wikipedia see exactly what they've always been accustomed to seeing in books is not going to confuse them. I.e.
is universally standard. If dx is an infinitesimal increment of a vector quantity rather than a scalar, and f(x) is vector-valued, then one sometimes writes
meaning the dot-product. In the latter case, one should of course write it in that way. Michael Hardy ( talk) 15:14, 24 January 2008 (UTC)
what about an open domain of a specific integral while the integral is defined by the domain?
how big is the difference between the integral on a closed domain vs open domain? 132.72.45.190 ( talk) 14:58, 15 January 2008 (UTC)
Dear Anonymous,
The difference between the integral on U and the integral on the closure of U may be arbitrarily large. See Smith-Volterra-Cantor set for the reason.
Sincerely,
Loisel ( talk) 17:52, 23 January 2008 (UTC)
This concept is not discussed in the article. I met it in Terence Tao's article on differential forms. —Preceding unsigned comment added by Randomblue ( talk • contribs) 19:00, 9 February 2008 (UTC)
shouldn't this be merged with Antiderivative? Professor Calculus ( talk) 00:30, 16 March 2008 (UTC)
It's the Calculus, not Calculus when referring to the subject of the Calculus. Every instance of the word calculus in this article would seem better if the article the preceded the word Calculus.
Yes, talk to some English major for confirmation of proper English usage in reference to the word calculus.
Mergatroidal (
talk)
23:54, 22 March 2008 (UTC)
The Calculus refers to the entire set of mathematical systems employed to determine change. And then there can be reference to a specific branch of the Calculus: the calculus of derivatives, the calculus of integrals, etc. I suppose it's easier to lay back and say who cares and drop the superfluous words, but if one wants to project the character of being precise, of being more precise with their words (hey professor, this is you I'm talking to ...!), wouldn't it appear appropriate in a classroom, let's say, to refer to the Calculus in this way? All these opinions on proper usage and yet no one knows what's correct? Hogwash. Put your foot down and be the first. Stick out (like a sore thumb.) Think different. Do it. (:~}
Would you be so cavalier in speech to such a stellar mind as this? Mergatroidal ( talk) 01:55, 25 March 2008 (UTC)
Lambiam, your discourse convinced me. In common usage among those who find the calculus just another one of oh so many mundane things that exist in life, and if one is not inclined to put on airs to the average man, it would seem stuffy, pedantic to employ the article the when referring to the calculus. Though if one were to mingle with other fans of mathematics, let's say at a party with other mathematicians, slipping in the article in conversation when casually referring to the calculus would seem appropriate and understood to convey the personal esthetic sense of appreciation one has for the beauty of what the calculus is all about. Personally I have recognized total stranger's perceptions for the grandeur of what the calculus is all about when they used the article of description in conversation, and isn't this sense what words are supposed to do: to convey one's thoughts and feelings? It's all context, and attitude. The common Wikipedia author speaks to the common man. Most Wikipedia authors are not elitist, and that is a bit of unintended sarcasm. I suppose let's not strive for the ideal inside Wikipedia articles, and appear snobby, elitist.
"There goes the King!" or "There goes that King guy." Mergatroidal ( talk) 22:17, 12 April 2008 (UTC)
— While driving a taxi in New York, and I believe the conversation was about the history of mathematics, and from the back seat a jocular, "... the calculus" was uttered, and not so much to correct my second or third instance of use of the word calculus, instead the passenger was intending to impress upon me that the calculus is a remarkable accomplishment of the human mind. Other than what the Greeks accomplished, the calculus could be put on a pedestal. I am not that versed in mathematics, though I wonder if Silly rabbit could name five other mathematical accomplishments as remarkable, and on par with the calculus? The practicality of which is par excellence. Mergatroidal ( talk) 23:25, 21 April 2008 (UTC)
The fundamental theorem of calculus guarantees that once an antiderivative is known, a definite integral can be computed. I therefore see no need for this edit, and find the weasel words unnecessary and misleading. I'm willing to be swayed by a detailed and convincing rationale for the edit. siℓℓy rabbit ( talk) 00:53, 24 May 2008 (UTC)
59.103.25.113 ( talk) 13:30, 15 August 2008 (UTC) By. Asad S. Yousaf Dated 15th Aug 2008 I was asked to place links to my Area Applets on Talk page rather than on Main page for Integrals. As definite integrals are used to approximate Area under and between curves, Volumes of Solids of Rotation, Length of a Plane Curves, Surface Area, Centroid, and so on. I had implemented online demonstration Java applets that illustrate the concepts just cited. So far I have developed 20 applets, and more can be expected. Yet I realized, putting up a demonstration program on web page without elaboration of the underlying concept served no purpose. So my applet pages are being reworked to include discussion of the topic along with presentation of the applet. So far Area under and between curves and Arc Length pages have been updated. Since Wiki is home to many Calculus topics, I thought your viewers may find my tools useful if they can interact with them to have a Visual representation of such concepts. Allow me to mention the the links to Area applet pages. Area under a Curve Applet is viewable at [6] Area between Curves Applet is viewable at [7] I am all ears to your feedback
59.103.25.176 ( talk) 15:57, 15 August 2008 (UTC)Asad S. Yousaf I have fixed the first two problems you pointed out.
59.103.27.86 ( talk) 18:26, 15 August 2008 (UTC)Asad S. Yousaf
sqrt(100^2-x^2)
for x
from -100
to 100
is given as Arc Length = 311.8
instead of 314.16 (100π). --
Lambiam
09:19, 20 August 2008 (UTC)In the introduction, I just made a change:
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'm concerned about some of the notation in this article. It's in the Introduction section.
I've never seen 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)
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:
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)
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).
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 ∫x1/2dx = 2/3*x3/2? MathMaven ( talk) 16:10, 6 March 2010 (UTC)
A request for comments has been filed concerning the conduct of Jagged 85 ( talk · contribs). 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)
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)
Close per WP:NOT#FORUM |
---|
The following discussion has been closed. Please do not modify it. |
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. [1] this:
∫√(1-e²sin²θ)dθ can anyone help me???@@@Thanks. 218.102.106.24 ( talk) 14:30, 7 July 2010 (UTC) |
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)
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:
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)
The area of a region is increasing by a rate of : 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 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)
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 , 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 have the properties as above. Is "the area under the graph of between and " not a valid, formal definition of ? If not, why not?
Cheers! Trolle3000 [talk] 05:32, 23 June 2011 (UTC)
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 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:An integral is a mathematical object that can be interpreted as an area or a generalization of area.
What do you say? Trolle3000 [talk] 17:27, 23 June 2011 (UTC)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.
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. [2] verification needed
- ^ http://wood.mendelu.cz/math/maw/integral/integral.php
- ^ Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3): 163-174 [165]
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)
"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 of cos x/sin²x — Preceding unsigned comment added by 41.221.159.84 ( talk) 15:58, 18 February 2012 (UTC)
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)
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)
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)
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)
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)
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)
This is an archive of past discussions. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page. |
Archive 1 | Archive 2 | Archive 3 | Archive 4 | Archive 5 |
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)
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)
In the lead of this article, the history is summarized with poor wording. According to the "History" section, the concept of integration was not "formulated by" Newton and Leibniz. They formulated the concept as we know it today, using for the first time a "systematic approach". They "generalized" and "formalized" it in a way, but the concept itself, in a less systematic and less general form, was formulated centuries earlier, as written in the History section. I tried to improve, but someone very unpolitely just reverted my edit without explaining nor trying to understand my rationale, which I explained clearly in my edit summary... Paolo.dL 07:46, 10 August 2007 (UTC)
Thank you, your edit improved the sentence. I understand and accept your rationale. Paolo.dL 21:16, 10 August 2007 (UTC)
Forgive me if I'm wrong, but wasn't the fundamental theorem of calculus (which by the way, was better-written and more accurate than the one on that page itself) specifically requested for this article? I spent a while getting a completely correct statement down as generally requested. I understand that the article is long, but having an article on integration without this was deemed incomplete. And now it's no longer there. Thoughts? Xantharius 15:43, 9 August 2007 (UTC)
Now that the article has stabilized a little, perhaps it's time to think about the Introduction. One of the first things that I feel should happen is the elimination of the plural first person and the conversational tone, as per the Wikipedia Mathematical MOS. This can be a hard thing to do, though. Also, I'm not sure about being asked to consider a swimming pool. What if I don't want to? (Half-joking.) Further, this is a good analogy, perhaps, but swimming pools often just have linear slopes, which therefore don't require calculus to find their volumes.
Perhaps we can get some consensus before going to town on this, and then do and evaluate. Xantharius 18:48, 10 August 2007 (UTC)
Why is the introduction restricting line integrals to two to three dimensional functions? Rubybrian ( talk) 14:19, 10 December 2007 (UTC)
Integral is in my view getting quite close to successful WP:GA nomination. The coverage of topics is (at least) sufficient at the moment, and the lead is certainly among the best-developed and accessible among maths articles.
However, I feel there are still certain issues with the overall structure and the flow of the article, issues that have to do with the organisation of mathematical content. I also feel some of these issues would be easier to resolve if the more specialised articles referenced from / referencing to Integral were in better shape and formed a more coherent whole. I have added a preliminary proposal for work on these relating articles to better enable Integral to stay focussed.
I have collected below some obervations and proposals specific to the article Integral itself.
The article mixes different types of integral in a way which is likely confusing to a beginner. There are in a way two broad topics within the scope of the article:
As it stands, the article introduces (without much discussion) the general measure-theoretic integral (calling it Lebesgue integral) quite early. However, after that, the unifying nature (among the topics of type 1 above) of the general integral is largely lost, and various different integrals (Riemann-Stieltjes, Lebesgue-Stieltjes, Daniell) are introduced/mentioned without making the connections clear. One thing I'm afraid can easily happen is that the (general) Lebesgue integral gets confused with integration with respect to the Lebesgue measure. In similar vein, multiple integral (Fubini's theorem, really) is introduced as a way to move beyond integrands defined on more general domains than intervals in R, despite the fact that the genereal integral has already been defined. The "Properties of integral" section discusses linearity three times in increasing generality, then discusses various inequalities that hold for the general integral in terms of the Riemann integral. Finally, the "Extensions" section introduces three quite different concepts on the same level: improper integrals, multiple integrals and (the various types of) integrals of differential forms. Multiple integrals should probably appear earlier, while the integrals of differential forms should really be its own top-level section (preferably a short one that points to another article).
"2 Terminology and notation" and "5.3 Conventions" should probably be integrated with other sections, and Conventions made much shorter.
The History section should cite a few sources. Good ones (also for some expansion) are the Historical Notes in Bourbaki's Functions of a Real Variable (pp. 129-162 of the English edition) and Integration (pp. V.123-136). History should probably be moved later in the article.
Most of what I would suggest doing to the article has to do with restructuring and moving text around, condencing where possible (hopefully leveraging related articles brought in line with this one). Stca74 13:09, 9 September 2007 (UTC)
Question?? 1)why we are using integration??can you give real time examples?? 2)where does it come to the picture?? —Preceding unsigned comment added by 60.254.13.8 ( talk) 18:21, 19 September 2007 (UTC)
Skand swarup ( talk) 13:44, 8 May 2008 (UTC) Integration is used to find areas of figures which are not geometric. Suppose you spill water on the floor and want to find out what area the water has covered, you can do so by integrtion. What it does is that it breaks up the non-geometric shape into a number of tiny geometric shapes. It then calculates the area of each of the tiny figures and adds them up. This of course gives only an approximation to the actual area.
Leland McInnes reverted my attempt to make the article more accessible with the comment
With respect I believe you need to offer more of an explanation. The text I appended to the intro paragraph was the following.
To be frank the article as it stands is a bit "gear-headed". That is, mostly the only people who would understand it would be those who already have a background in calculus. In principle since an encyclopedia is supposed to be generally accessible (Wikipedia even more than most) it is worth at least trying to make the introduction clarify the subject matter for as wide an audience as possible (I'd argue that the goal should be broader than that but at least that is a starting point).
-- Mcorazao 21:52, 12 October 2007 (UTC)
Please comment or I will simply put back my edits. -- Mcorazao 15:15, 16 October 2007 (UTC)
I was making links for my page direct integration of a beam and when I was putting in the link for this page, I noticed that the lead for the article reinforces the perception that an integral only represents "the area under a curve." Since there's not another page on mathematical integration, this is more of an issue in this article. I would think the lead should point out to readers that what f(x)dx represents is a box with width dx and height f(x), and that the integral of it is a summation of these rectangles' areas as dx->0 while x goes from a to b. JW 05:44, 13 October 2007 (UTC)
seeing the wide variety of meanings at integral (disambiguation), and the status of the word integral as a generic English word, I wonder whether it wouldn't be advisable to move this article to integral (calculus) on grounds of the "principle of least surprise". dab (�) 18:21, 13 November 2007 (UTC)
A tag has been placed on the article. I am tempted to just remove it whilst citing Wikipedia:Scientific citation guidelines, but I think that a discussion should take place here first for the sake of completeness. Thoughts? — Cronholm 144 05:39, 13 December 2007 (UTC)
It seems to me that the animation associated with the line integral section is incorrect or at least misleading (I feel incorrect). It does give a good graphical explanation of a mathematical operation; however, the operation shown is not a line integral. As I understand it, a line integral on a vector field returns a scalar value; the 'sum' of the dot products between the unit tangent vector to the curve and the field value at each point. However, the graphic shown just gives the 'sum' of the field values (not dot products) (and is vector valued).
I am only a second year degree student, so it is quite possible there are other definitions of line integration I don't know of, or that I am just incorrect. However, being a degree student I have been directly affected by the animation; not knowing the definition of a line integral, I used the animation as a definition in a set problem (which caused problems for me!). I can say from experience therefore that the animation was misleading and definately unhelpful. If it is a (rather than the) correct definition, a note to this effect would be very useful.
If I don't find a reply or changes to the article, I will remove the image to avoid other people getting the same problems from it I did. I'm also posting this message in the main Line Integral article, which uses the same image. —Preceding unsigned comment added by 88.106.245.46 ( talk) 12:06, 25 December 2007 (UTC)
I don't think that this particular animation adds anything of value. On the other hand, yes, it is very confusing, and should better be removed. Having said that, I am truly amazed that anyone would try to infer a basic mathematical definition from an illustration in the middle of a wikipedia article! Arcfrk ( talk) 17:59, 29 December 2007 (UTC)
This article reads too much like a text book for trained mathematicians. It is largely inaccessible to lay-people in search of general and simplified knowledge of the subject. I found it disappointingly unhelpful. 143.97.2.35 ( talk) 16:07, 27 December 2007 (UTC)
What were you hoping for? Why do you want to know about integration? The article does need some mathematical knowledge to understand, but there is a limit to how simple it can be reasonably made. Do you need to know what integration in general is, how to do it, or about a specific type of integration (line integration for example)? If you are more specific I will try to improve the article.
The juxtaposition of two symbols have different meanings.
This is hard to newcomers. I suggest that the multiplication sign be written explicitely in order to reduce the confusion.
Write
rather than
Any objections? Bo Jacoby ( talk) 22:55, 9 January 2008 (UTC)
Dear Bo,
I'm not sure that I like the idea of because I think it might mislead a reader into thinking that is a number. It is useful to emphasize that is not a product of and , which is why one usually puts a \, between the and , like so:
Loisel ( talk) 03:34, 10 January 2008 (UTC)
I think it is a poor idea, since interpreting f(x)dx as the product of f(x) and dx isn't necessarily correct and might lead one to believe dx actually represented an infintesimal, which is certainly quite wrong for many (or indeed almost all) defintions of integral. Ultimately for many definitions the dx is purely formal notation and not really representative of anything. On those grounds I suggest that "making the multiplication explicit" is actually more misleading than helpful. -- Leland McInnes ( talk) 17:07, 10 January 2008 (UTC)
Well, ladies and/or gentlemen, notational convenience is the same thing as algebra, and there is nothing wrong in that. I was surprised that the disagreement was about the multiplication itself, rather than about the multiplication sign. The simpler introduction to differentiation and integration is to begin with polynomials. If x is a formal variable, then so is dx. The rules of algebra are d(x+y)=dx+dy and d(x·y)= x·dy+dx·y. These rules are sufficient for deriving the rules for differentiating a formal power series and for solving differential equations. The rules of interpretation is that if dx is not zero, then x is neither constant, nor maximum, nor minimum. The interpretation in terms of limits, and the Riemann and Lebesque integrals, are not needed for quite a while. The beginner needs a break. Wikipedia should explain, rather than just repeat unintelligibly advanced stuff. Bo Jacoby ( talk) 15:13, 20 January 2008 (UTC).
The omission of an explicite multiplication sign is widespread. That does not mean that the multiplication sign is a neologism. The article on polynomial omits the multiplication sign, but that does not mean that multiplication is not intended, for example 2xy2 means 2·x·y2. Don't you all agree on that? Omitting the multiplication sign makes no harm until juxtaposition means something else than multiplication. Then confusion appears. For example (f+g)(x) = f(x)+g(x) defines the sum of two functions, f and g. Here the juxtaposition (f+g)(x) does not mean the multiplication (f+g)·(x). Mathematicians don't mind very much, because the parenthesis around (x) indicate that x is argument to a function, but parentheses have other meanings. In the expression (f+g)(x+y) it is less clear whether the interpretation f(x+y)+g(x+y) or (f+g)·(x+y) is intended. Have for example a look on the articles catenary and gamma function and identify which juxtapositions in the formulas indicate multiplications and which ones do not, and why. Explicite multiplication signs sure would help a lot. So, omission of multiplication signs in formulas may be polite to the author, but it is rude to the reader. Regarding integration there is no doubt that Leibnitz intended a multiplication between the function value f(x) and the differential dx: The differential of the area bounded by the x-axis, the y-axis, the curve y=f(x), and the vertical line at x, is the height f(x) times the base dx. The difficulties in interpreting the differentials have historically lead to tricky definitions for derivation and integration, but the algebraic axiomatic approach avoids these complications. You do not need to know what a differential is, as long as you can use it correctly in computations, just as you do not need to know what −3 means, except that is solves the equation x+3=0. Bo Jacoby ( talk) 02:13, 22 January 2008 (UTC).
Dear Lenand McInnes. Yes, we are talking formal infinitesimals. No, I am not defining my own integral. If
does not mean
then you cannot deduce that
because then you cannot rely on the distributive rule of multiplication:
In the expression for the Riemann sum
multiplication is obviously implied
Also the article Darboux integral defines the upper Darboux sum of ƒ with respect to P:
Here too juxtaposition means multiplication:
Mainstream litterature on Riemann integral, Lebesgue integral and Henstock-Kurzweil integral, as well as the corresponding wikipedia articles, generalizes the elementary Leibnitz integral assuming that the readers are already familiar with elementary high-school algebraic integration. This assumption cannot be made here. I thought that the confusion about the interpretation of juxtaposition was confined to beginners, but now I realize that I was wrong. Bo Jacoby ( talk) 11:32, 22 January 2008 (UTC).
I'm just curious. What does the juxtaposition of f(x) and dx in "f(x)dx" mean if it does not mean multiplication? Is the formula d(2x)=2dx not involving two multiplications? Bo Jacoby ( talk) 15:47, 22 January 2008 (UTC).
Thank you for answering. I understand that you consider "d(2x) = 2dx" illegitimate while "d(2x)/dt = 2dx/dt" is legitimate, meaning "d(2·x)/dt=2·dx/dt", involving two multiplications. Am I right? "∫ dx = ∫ 1dx". Right? "∫ f(x)dx = ∫ (f(x)·1)dx = ∫ f(x)·1dx = ∫ f(x)·dx". Right? Algebraic shortcuts among friends seems to be taboo i WP even if they work. The Riemann integral was supposed to generalize the antiderivative, rather than to restrict the algebraic freedom. Bo Jacoby ( talk) 13:56, 23 January 2008 (UTC).
The argument used in Fundamental_theorem_of_calculus#Intuition for writing meaning does perhaps involve handwaving, but the differential algebra does not. There is no handwaving about the rules of differentiation. Differentiation d is defined on the polynomial ring Q[x] by introducing another variable, dx, and the rules that d(x)=dx, and that dk=0 when k is a number, and d(X+Y)=dX+dY, and d(X·Y)=dX·Y+X·dY when X and Y are polynomials. This is sufficient in order to specify formal differentiation on power series, and all the entries in lists of integrals follow. Limits are not needed for the algebra, only for the interpretation. This means that d(2·x)=2·dx is an algebraic fact. It does not rely on Riemann integration. Bo Jacoby ( talk) 16:54, 23 January 2008 (UTC). PS. The lists of integrals says: Are you saying that this elementary formula doesn't represent correct mathematics? I politely request you to tell me which ones of my statements above, that end with "right?" or "am I right?", that you consider wrong. The summary statistics, that most of them are wrong, is not sufficient. I am trying to figure out what you guys mean. Using your (nonstandard) notation Ix(f(x)), I note the rule that Iax(f(x))=Ix(af(x)), showing that a constant factor can be moved between two arguments, which implies multiplication. Bo Jacoby ( talk) 11:52, 24 January 2008 (UTC).
Letting new readers of Wikipedia see exactly what they've always been accustomed to seeing in books is not going to confuse them. I.e.
is universally standard. If dx is an infinitesimal increment of a vector quantity rather than a scalar, and f(x) is vector-valued, then one sometimes writes
meaning the dot-product. In the latter case, one should of course write it in that way. Michael Hardy ( talk) 15:14, 24 January 2008 (UTC)
what about an open domain of a specific integral while the integral is defined by the domain?
how big is the difference between the integral on a closed domain vs open domain? 132.72.45.190 ( talk) 14:58, 15 January 2008 (UTC)
Dear Anonymous,
The difference between the integral on U and the integral on the closure of U may be arbitrarily large. See Smith-Volterra-Cantor set for the reason.
Sincerely,
Loisel ( talk) 17:52, 23 January 2008 (UTC)
This concept is not discussed in the article. I met it in Terence Tao's article on differential forms. —Preceding unsigned comment added by Randomblue ( talk • contribs) 19:00, 9 February 2008 (UTC)
shouldn't this be merged with Antiderivative? Professor Calculus ( talk) 00:30, 16 March 2008 (UTC)
It's the Calculus, not Calculus when referring to the subject of the Calculus. Every instance of the word calculus in this article would seem better if the article the preceded the word Calculus.
Yes, talk to some English major for confirmation of proper English usage in reference to the word calculus.
Mergatroidal (
talk)
23:54, 22 March 2008 (UTC)
The Calculus refers to the entire set of mathematical systems employed to determine change. And then there can be reference to a specific branch of the Calculus: the calculus of derivatives, the calculus of integrals, etc. I suppose it's easier to lay back and say who cares and drop the superfluous words, but if one wants to project the character of being precise, of being more precise with their words (hey professor, this is you I'm talking to ...!), wouldn't it appear appropriate in a classroom, let's say, to refer to the Calculus in this way? All these opinions on proper usage and yet no one knows what's correct? Hogwash. Put your foot down and be the first. Stick out (like a sore thumb.) Think different. Do it. (:~}
Would you be so cavalier in speech to such a stellar mind as this? Mergatroidal ( talk) 01:55, 25 March 2008 (UTC)
Lambiam, your discourse convinced me. In common usage among those who find the calculus just another one of oh so many mundane things that exist in life, and if one is not inclined to put on airs to the average man, it would seem stuffy, pedantic to employ the article the when referring to the calculus. Though if one were to mingle with other fans of mathematics, let's say at a party with other mathematicians, slipping in the article in conversation when casually referring to the calculus would seem appropriate and understood to convey the personal esthetic sense of appreciation one has for the beauty of what the calculus is all about. Personally I have recognized total stranger's perceptions for the grandeur of what the calculus is all about when they used the article of description in conversation, and isn't this sense what words are supposed to do: to convey one's thoughts and feelings? It's all context, and attitude. The common Wikipedia author speaks to the common man. Most Wikipedia authors are not elitist, and that is a bit of unintended sarcasm. I suppose let's not strive for the ideal inside Wikipedia articles, and appear snobby, elitist.
"There goes the King!" or "There goes that King guy." Mergatroidal ( talk) 22:17, 12 April 2008 (UTC)
— While driving a taxi in New York, and I believe the conversation was about the history of mathematics, and from the back seat a jocular, "... the calculus" was uttered, and not so much to correct my second or third instance of use of the word calculus, instead the passenger was intending to impress upon me that the calculus is a remarkable accomplishment of the human mind. Other than what the Greeks accomplished, the calculus could be put on a pedestal. I am not that versed in mathematics, though I wonder if Silly rabbit could name five other mathematical accomplishments as remarkable, and on par with the calculus? The practicality of which is par excellence. Mergatroidal ( talk) 23:25, 21 April 2008 (UTC)
The fundamental theorem of calculus guarantees that once an antiderivative is known, a definite integral can be computed. I therefore see no need for this edit, and find the weasel words unnecessary and misleading. I'm willing to be swayed by a detailed and convincing rationale for the edit. siℓℓy rabbit ( talk) 00:53, 24 May 2008 (UTC)
59.103.25.113 ( talk) 13:30, 15 August 2008 (UTC) By. Asad S. Yousaf Dated 15th Aug 2008 I was asked to place links to my Area Applets on Talk page rather than on Main page for Integrals. As definite integrals are used to approximate Area under and between curves, Volumes of Solids of Rotation, Length of a Plane Curves, Surface Area, Centroid, and so on. I had implemented online demonstration Java applets that illustrate the concepts just cited. So far I have developed 20 applets, and more can be expected. Yet I realized, putting up a demonstration program on web page without elaboration of the underlying concept served no purpose. So my applet pages are being reworked to include discussion of the topic along with presentation of the applet. So far Area under and between curves and Arc Length pages have been updated. Since Wiki is home to many Calculus topics, I thought your viewers may find my tools useful if they can interact with them to have a Visual representation of such concepts. Allow me to mention the the links to Area applet pages. Area under a Curve Applet is viewable at [6] Area between Curves Applet is viewable at [7] I am all ears to your feedback
59.103.25.176 ( talk) 15:57, 15 August 2008 (UTC)Asad S. Yousaf I have fixed the first two problems you pointed out.
59.103.27.86 ( talk) 18:26, 15 August 2008 (UTC)Asad S. Yousaf
sqrt(100^2-x^2)
for x
from -100
to 100
is given as Arc Length = 311.8
instead of 314.16 (100π). --
Lambiam
09:19, 20 August 2008 (UTC)In the introduction, I just made a change:
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'm concerned about some of the notation in this article. It's in the Introduction section.
I've never seen 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)
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:
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)
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).
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 ∫x1/2dx = 2/3*x3/2? MathMaven ( talk) 16:10, 6 March 2010 (UTC)
A request for comments has been filed concerning the conduct of Jagged 85 ( talk · contribs). 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)
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)
Close per WP:NOT#FORUM |
---|
The following discussion has been closed. Please do not modify it. |
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. [1] this:
∫√(1-e²sin²θ)dθ can anyone help me???@@@Thanks. 218.102.106.24 ( talk) 14:30, 7 July 2010 (UTC) |
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)
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:
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)
The area of a region is increasing by a rate of : 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 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)
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 , 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 have the properties as above. Is "the area under the graph of between and " not a valid, formal definition of ? If not, why not?
Cheers! Trolle3000 [talk] 05:32, 23 June 2011 (UTC)
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 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:An integral is a mathematical object that can be interpreted as an area or a generalization of area.
What do you say? Trolle3000 [talk] 17:27, 23 June 2011 (UTC)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.
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. [2] verification needed
- ^ http://wood.mendelu.cz/math/maw/integral/integral.php
- ^ Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3): 163-174 [165]
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)
"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 of cos x/sin²x — Preceding unsigned comment added by 41.221.159.84 ( talk) 15:58, 18 February 2012 (UTC)
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)
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)
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)
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)
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)
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)