Talk:Integral/Archive 2

message: Substituted at 14:32, 14 April 2016 (UTC)
thread_title: Assessment comment

message: Shouldn't we include a proof that integrals can be calculated through antidifferentialtion?
I'll see if I can write one —The preceding unsigned comment was added by Sav chris13 (talk • contribs) 05:10, 5 February 2007 (UTC)
thread_title:  Antiderivatives 

message: Let f(x) be a function 
and F(x) [the integral] be the function of the area under f(x)  
Also note the relationships:
    Area = Length X Width
    Gradient = Rise / Run
The Rise in F(x) is F(x + h) - F(x)    as h --> 0
The Run is h
The derivative of F being repressented by 
    [F(x + h) - F(x)]/h    as h --> 0
Now F(x) is the Area function
And values along the x-axis represent the "width" of our area
So h is the width of this area
So 
    Gradient = Rise / Run = Area / Width = Length
Ie:    [F(x + h) - F(x)]/h = f(c)
For some value of c which is between (x + h) and x
Now as h --> 0     c will approach x
    [F(x + h) - F(x)]/h = f(x)  as h --> 0
Hence the relationship between F and f is
    F'(x) = f(x) —The preceding unsigned comment was added by Sav chris13 (talk • contribs) 07:19, 5 February 2007 (UTC)
replies: text: If you wish to help with this month's collaboration, please make your edits (and comments) at the sandbox version. Thanks. --KSmrqT 15:18, 9 June 2007 (UTC)
thread_title:  Proof integral is anti-derivative 

message: I think we should consider including the definition of the integral in this article. That is, 
the lim N->infinity of the summation from k=1 to N of f(a-kΔx)Δx, where Δx=(b-a)/N. This is describing the method of using an (approaching) infinite number of rectangles to produce the area of the function f in the interval [a,b].
This is in relation to integration on 2D planes. I am well aware that there are refined definitions for integrals of other circumstances, which should also be included. The reason I bring this up is because we include the definition of the derivative under its own section but only give the fundamental theorem of calculus under the Integration section. I feel this definition should be included.
If we could, I would like to discuss this and if others want, I could spearhead the initiative myself(of course, with the help of others).
Gagueci 20:39, 1 May 2007 (UTC)
replies: text: Please do not post the same comment multiple times, it may be considered vandalism. I'm also in favour of including two brief sections on the two most popular definition of the integral here. Loom91 07:03, 2 May 2007 (UTC)
thread_title:  Definition of Integral 

message: As outlined in my rating comment, I think the scope of this article needs to be broadened to cover the concept of integral in appropriate generality, not concentrating only on integrals of real-valued functions of one real variable. While this is a critical special case (and indeed the key building block for other inegrals), it is by no means sufficient for an article that aims to cover one of the most critically important mathematics concepts.
However, care should be taken not to introduce better coverage at the expense of too high level of demands for readers — this is likely one of the most viewed maths articles. Therefore most technical details belong to either in later sections or in particular separate articles. 
Additions and changes proposed include:
* Add multiple integrals or integrals of functions of several variables (a truly fundamental concept)
* Mention integrals of vector-valued functions (less elementary but important) and provide link
* Mention briefly integration of differential forms (and its somewhat more elementary version as countour integrals) and provide
* Better explanation of integral as a linear operator; its continuity properties
* More elaboration of integral as a (weighted) average; relation to expected value in probability / statistics
* Better references to Lebesgue integral and measure theory;
* Better pointers to applications;
* More focus in the list of various integrals: Lebesgue and one "simple" approach (Riemann or Daniell) + Stieltjes version should probably suffice for topics to be discussed in the article; pointers to other approaches could be made less prominently without bullet point lists;
* History of integral should be explained 
* The lead should be compacted quite a bit and material moved to actual article sections.
:Stca74 17:39, 15 May 2007 (UTC)
replies: text: A very thoughtful and detailed suggestion. I agree completely on all points except the last one. I had proposed some of those before. I will further like to propose very brief discussions of the Rieman and Lebesgue integral in the article. Loom91 17:45, 15 May 2007 (UTC)
replies: text: Agree as well. Integral is a Math colabortation of the week candidate. If it gets that it might get some more attention. --Salix alba (talk) 18:18, 15 May 2007 (UTC)
thread_title:  Broader coverage needed 

message: In general a good article, but two points:
* Is that Arabic integral symbol a joke?
* Line integrals are never mentioned.
Dmbrown00 04:31, 31 May 2007 (UTC)
thread_title:  2¢ 

message: This section must be elaborated on mathematically. Gagueci 19:11, 31 May 2007 (UTC)
replies: text: More detail can be found in the numerical integration article. Oleg Alexandrov (talk) 01:28, 1 June 2007 (UTC)
thread_title:  Approximation of Definite Integrals section 

message: Since the first paragraph (indeed, the first sentence) frightens me, and the rest of the article needs massive work as well, I have concentrated my initial efforts on non-creative writing. Specifically, I have laboriously searched the Web and added some great references for the "History of integral notation" section. In doing so, I have established a precedent that I wish to be followed: Harvard style references with automatic links. I have yet to properly templatize (!) the Leibniz citation, but that is a minor issue, which I will fix Real Soon. (I plead fatigue.)
I'm not yet concerned with naming names within the bulk of the article, because I expect that to happen as it is pummeled into submission. I see a need for better coverage of the basics (linearity!), but we should also touch on contour integrals, complex integration, measure theory and analysis, differential forms, and (if we're really brave/foolhardy) de Rham cohomology or some such. --KSmrqT 12:29, 1 June 2007 (UTC)
replies: text: In calculus, the integral of a function extends the concept of a sum over a discrete range to sums over continuous ranges. The process of evaluating (or determining) an integral is known as integration. Integration is often used to find the "total amount" of a property on some bounded domain, whenever that property varies in a smooth or continuous fashion. For example: finding the temperature or electric field strength of a volume of space, since both temperature and field strength vary continuously in space; or finding the total acceleration or velocity over an interval of time, since both vary continuously in time. However, the mathematical treatment of integration is sufficiently general that it can be used to work with with any property that can be viewed as undergoing variation over a continuous domain.
I still feel it is a little vague, and I'm dicing with the issue of smooth vs. continuous (smooth, in a general sense, is more accessible to laymen, but it has specific mathematical connotations which may be worth avoiding here). Suggestions and feedback are welcome. -- Leland McInnes 00:06, 3 June 2007 (UTC)
thread_title:  Drone work 

message: I noticed that there is a sentence in the article that mentions that it is possible to prove that x^x has no elementary antiderivative.  While I'm sure that's true, the way the sentence reads begs the question of how to go about proving it.  So it seems like there really ought to be a footnote citation that leads interested readers to an actual proof.  (It may even be included in one of the references at the end of the article, but if so there's no indication which part of which reference to go to for the proof.) Dugwiki 20:51, 1 June 2007 (UTC)
replies: text: While I'm sure it's true that a reference would be nice, this is an incredibly blind and superficial observation. Please read the comment just before yours. The only references in the article at present are the ones I added in the history of notation section — and we are using Harvard references, not footnotes. You propose putting lipstick on a pig; what the article needs most is a lot of good writing, with sparse, appropriate references added as we go. That is, if anyone is interested. --KSmrqT 01:19, 2 June 2007 (UTC)
thread_title:  Should include footnote citation to proof regarding x^x 

message: Loom91 removed my introduction, citing the fact that a function does not need to be continuous to be integrable. I agree, but then that is not what my introduction said -- the claim is that the function should be defined on a continuous domain since a function defined on a discrete domain can simply be summed over a subset of the domain in the usual manner. Integration deals with the issue of extending this to continuous domains, and I feel this provides the most natural "intuitive" description of what integration achieves. I would hope that we could discuss the introduction instead of just reverting it -- if nothing else it is better than the existing introduction which makes almost no sense to anyone who isn't well schooled in what integration is already. -- Leland McInnes 21:07, 3 June 2007 (UTC)
thread_title:  Discrete vs. Continuous 

message: Hey all, I have created a sandbox skeleton for the new look of the article and Leland has kindly outlined his vision for the article on the talk page. Please direct all major edits that you wish to try before adding them into the article proper there. Hopefully the skeleton can be given flesh in the sandbox and then life in the article proper. Cheers--Cronholm144 19:25, 4 June 2007 (UTC)
replies: text: I'll add my voice here: the article will benefit from a broad overhaul, but there's no point leaving the page a mess while we do it. I've now put the outline in the sandbox skeleton, so please come and have a look -- there is plenty of material to be filled in; from history, such as early Greek efforts at integration, through to robust formal definitions of Riemann and Lebesgue integrals, and discussions of line and surface integrals, differential forms, and de Rahm cohomology. If all the collab of the month people can drop by the sandbox version and help fill out their area of expertise we can end up with a great article by the end of the month. -- Leland McInnes 19:35, 5 June 2007 (UTC)
thread_title:  Sandbox Skeleton