Talk:Integral/Archive 1

This is an archive of past discussions about Integral. 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

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Archive 5

Initial comments

I have removed this:

Integration, however, is particularly hard for computerized algebra systems. Although newer systems have improved, even the best systems are not nearly as effective as an experienced human.

The first time I read it I was a bit surprised, but then decided the person who wrote it must know something I don't. However, when I accidentally read it again today, I decided it either needs to be supported by some hard data, or be removed. In my experience, Maple is very good at finding antiderivatives. The Risch-Norman algorithm is very general and efficient. See On the Risch-Norman integration method and its implementation in MAPLE by Geddes and Stefanus.

I have replaced the above statement with a generic note about how it's hard to find antiderivatives.

If you're talking about certain special definite integrals which can be solved by residue calculus (say), I've also had good experience integrating those in Maple.

Please clarify if you want to restore this text. Loisel 19:28, 21 May 2004 (UTC)

Maybe we can make this statement more precise. I believe it's uncontroversial to say that integration is hard because there is no "one size fits all" algorithm. The current revision says as much. Can we make this more precise by enumerating mentioning some general symbolic integration algorithms, and mentioning what they don't cover? -- FWIW, I've tried to solve integrals, arising in Bayesian statistical inference, using Mathematica, and as often as not Mathematica can't find a symbolic solution. So I'm inclined to think that symbolic integration is still hard. Regards & happy editing, Wile E. Heresiarch 02:00, 23 May 2004 (UTC)

It would be fantastic if you could give the specific example, and even better if you could give a hint as to why the standard algorithms don't work. If the integral was indefinite, we'd need to have the antiderivative to show that we're better than the computers. Loisel 11:09, 1 Jun 2004 (UTC)

Well, here are some examples. I used the Mathematica integrator web interface, which can find indefinite integrals but not definite integrals. Some of these problems are more usefully stated as definite integrals but I didn't try that. Mathematica just returned the integral for all of these four.

(1) Exp [-(y-x)^2] BesselK [0, Abs[x]]

(2) Exp [-(y-x)^2] Exp [-Log [x]^2]/x

(3) x Exp[-x^2/2] (1/2) (1+Erf[x/Sqrt[2]])^2

(4) Exp[-x^2]/(Exp[-x^2]+Exp[-(x-3)^2]) Exp[-(v-x)^2] Exp[-(x-5)^2]

Nos. 1 and 2 originated from finding the sum of two variables with different distributions, so those are convolutions. No. 3 is the expected value of the largest of three Gaussian variables. No. 4 came up in trying to find a marginal distribution in a case where some conditional distributions were mixtures of conditional Gaussians. -- I don't know the solutions for any of these, which is why I was trying Mathematica. I can't claim to be better than Mathematica; however, what is of interest here is that Mathematica is no better than me, for these four problems. Well, I suppose I might have entered the commands wrong or something; if so someone will soon straighten it out. Comments? Wile E. Heresiarch 04:42, 2 Jun 2004 (UTC)

Ah, I was hoping for an example where a closed-form primitive was available. I'm not completely certain, but I suspect the functions you offer do not have closed-form primitives. As you most probably know, the function exp(x^2) does not have a closed-form primitive either (and one can prove so.) Depending on your definition of "closed form", the Risch-Norman algorithm should always be able to integrate formulae which do have closed-form primitives. I'm not exactly sure what your background is, so in case you're not a mathematician, I'm saying that I think one could prove that it is impossible to write a formula whose derivative exp(x^2) (or one of the functions you give above.) While your functions definetly have antiderivatives, these antiderivatives will never be writable in a nice way. Hence, claiming that the computer is less good than people at antidifferentiating based on such evidence would be misleading. Loisel 07:22, 2 Jun 2004 (UTC)

I see what you're getting at, but I think it's beside the point. "Integration is hard in principle" is entirely consistent with "integration is hard for computer algebra systems". If there's no possible solution, then say so; Mma doesn't distinguish between "too hard for Mma" and "impossible", and that's definitely a shortcoming. Even in this case, there are avenues of attack, such as finding an expression containing simpler integrands, or attempting to construct some sequence of approximations -- those are things a human could try, and there's no particular reason algebra systems couldn't do it too. Getting back to the article, it would help to outline the scope of the integration problems that are known to have solutions and point out how easy it is to go beyond those boundaries. For what it's worth, Wile E. Heresiarch 15:36, 2 Jun 2004 (UTC)

On rereading the comment history, I see that you (Loisel) and I seem to be addressing different points. "Even the best systems are not nearly as effective as an experienced human" is an interesting statement, and might even be true, but I'm not concerned with defending it. To improve the article, I think we can steer away from that contentious assertion and just describe why integration is hard, what's possible, how much of the possible is now handled by computer systems, etc. Happy editing, Wile E. Heresiarch 16:14, 2 Jun 2004 (UTC)

Two comments about the current revision (2004/02/09). (1) A few days ago I put "The integral of a function is, roughly speaking, an area, mass, ..." which was modified to "...can be used to represent an area, mass, ...". I deliberately chose "is" because the article needs to say what the integral -is-. I'll propose reversion unless someone else has a better candidate to say what an integral -is-. (2) There is some discussion about Riemann and Lebesgue integrals. This replicates material found in the articles on Riemann and Lebesgue integrals, so I'm inclined to suggest it be cut back to a summary and reference to those other articles. Happy editing, Wile E. Heresiarch 03:16, 11 Feb 2004 (UTC)

I'm the one who wrote most of these articles (Riemann integral, Lebesgue integral and Integral.) The text in Integral does duplicate a bit of information, but the point in Integral was to be able to give a very coarse idea of how the two mainstream area-based theories of integration differ. If you want to remove this text, that's okay with me, but I still think that the nuance between the Riemann and Lebesgue integral should at least be outlined in this article. The reader should not have to be familiar with either integral in order to get a basic explanation of a few paragraphs. Loisel 19:33, 21 May 2004 (UTC)

I agree that the integral article ought to mention Riemann and Lebesgue. Maybe instead of mechanics, this article can outline why there is not a single definition of the integral, and then leave the details to the Lebesgue integral article. I'll give it a try in the next day or two and we can see how successful that is. -- To go back to point (1) above, I've attempted to state a definition using the word "is". I may not have been completely successful with what I wrote today ("In calculus, the integral of a function is a generalization of area, mass, volume, total, and average"), but if that's still off the mark, I'd like to suggest that it be replaced by something which likewise says what the integral -is-. FWIW, & happy editing, Wile E. Heresiarch 01:48, 23 May 2004 (UTC)

The notation for the floor function is incorrect - I'll look into this to see if it can be done more effectively. -- User:David Martland

Something like $\lfloor x\rfloor$ ? -- The Anome 07:52 25 Jun 2003 (UTC)

Or ⌊ x ⌋ ? - The Anome

New Subject:

When someone types in "integration" or "integrate", they get Wikipedia rules for page integration for integrate and Mathematics Integral for Integration. In technology, integrate means "to integrate to application or purpose" >" to create a set of interdependencies to make a primary function possible." An integration in technology is "A number of dissimilar systems or components interrelated and interdependent in such a manner as to make a primary function possible." Examples are the computer and the automobile. If one studies a chip on a motherboard or the brakes on an automobile as a standalone and not as an interdependency of integration, the interpretive lens will be disoriented and the understanding will be distorted. It follows that if one studies a process in the natural world, outside the integrated whole in which it resides and views it as a standalone and not as an interdependency of a much larger integration, the interpretive lens will be disoriented and the understanding and thus the attempted explanation will be distorted. This may lead one to build an entire thesis off of a tangent that does not exist.

An integration may be static or dynamic. It may be lateral or dynamically layered. The lens of Logic requires a lens of an enginner or architect or Information Specialist; preferably all three.

Interdisciplinary Study, connected learning and "thinking outside the box" are buzz terms used in the academic world, but so far have not been integrated and applied in the academic environment. A biochemistry professor should take a course in Information Science to better understand the "automation run from a genetic database" he [she] is studing.

The terms "integration" and "interdependencies of integration" turn a light on. They bring into focus an understanding that is hidden from view now. Integrations are information rich. But the information is embedded into the application or purpose. So without a view of the integrated whole, most of the information is invisible to the interpretive lens and not available for extraction. Integration brings a whole world to light.

Interpreting the word "integration," as abstractly as possible, it would appear that in one way or another it refers to the way in which distinct ideas or entities form a whole. The the differing values falling within the range of an integrand each play a part in the evaluation of the integration. While derivatives involve division of subtractions, integrals can be evaluated with a summation of multiplications. The word "integral" has been used in differing contexts by mathematicians: The integral can be seen as the the evaluation of the a whole antiderivation, ergo an integral symbol presented in conjunction with an integrand and the differential of the variable upon which the integrand depends; It can also be seen as the very integral symbol itself. The meaning of the integral is not heavily crucial to engineers who use the tool mathematics to implement, whereas those who define mathematics are a bit more philosophical, exempli gratia, Isaac Newton.

Personally, I do not have as high of a praise of the nomenclature of integration as I did once ago. I more inclined to used the following in works:

\left[\left({d \over dx}\right)^{-1}f(x)\right]_{u_{0}}^{u}

instead of the traditonal and accepted

\int _{u_{0}}^{u}f(x)\,dx

Lindberg G Williams Jr 04:35, 22 May 2004 (UTC)

Ah, today I finanlly stumbled upon fractional calculus which I have shared ideas similar though less developed. It is a concept I have never seen in a text book but which I have often mingled. Througout the course of my mathematical career I surmise that pregraduate scholars are not allowed to invent new ideas in math.

Lindberg G Williams Jr 17:09, 22 May 2004 (UTC)

Warning to Readers

This article must carry a warning (in the discussion area ONLY) that warns the gentle reader that reading this article will bring back the sweaty palms, sinking feeling in the stomach, blurriness in vision and the general feeling of being a total idiot, that was the Advanced Calculus class in college. Kudos to mathematicians. They are definitely a step higher in the IQ scale than the rest of us.

Use of symbols from set theory?

The symbols {,} and epsilon are used without definition or even reference to set theory. Some people may have had a lousy education and look towards this article for guidance. It would be swell if someone added that. Mbac 03:58, 6 March 2007 (UTC)

No mention of integration pertaining to simulation?

In computer graphics, when learning about physical simulations (e.g. particle systems, physically based animation), I see mention over and over again of integration as a key step in the process. Euler, Runge-Kutta and Verlet are all types of integration methods found in this context, but I'm having a hard time seeing the relationship between finding the area under a curve and calulating the coordinates for a moving particle at the next time step. Is it because both are dealing with discrete samples of a continuous value in some way?

I'd really appreciate it if someone could add a paragraph or two clarifying the relevance of integration to physics simulations a la computer animation and games.

Thanks!

Numerical integration is the name of that technique (which doesn't appear to be linked in this article, but should be). Essentially, to compute the motion of a particle, you are solving a differential equation, and in general this is done by performing an integration. Techniques like the Runge-Kutta method are ways of approximating this integration. -- DrBob 21:53, 16 Jun 2004 (UTC)

"Integration" is also used, by abuse of language, to discuss numerical ordinary differential equation solvers (they are called numerical integrators.) Euler is u_k+1=u_k+u'(x_k)dx. Runge-Kutta is a family of integrators that includes the Euler scheme, but also offers higher order (more precise) methods. Verlet (or Stormer-Verlet) is a second order scheme that preserves certain important quantities such as a Hamiltonian (similar to energy.) It is even symplectic (meaning that it preserves length, surface, volume, n-dimensional volume, etc...) Such schemes are said to be geometric, because they can be viewed as a discrete iteration that preserves certain geometric and physical properties of the continuous system. High order geometric integrators are very difficult to obtain, but are available. Strangely, over extremely long integration periods, geometric integrators with a fixed, non-adaptive step size are much more precise than non-geometric integrators (or geometric integrators with variable step size.) Precisely, when the truncation error becomes quadratic in the number of steps or worse for a traditional scheme, schemes such as Stormer-Verlet will still have mainly linear error in the number of steps. Independently of this, because geometric integrators preserve many useful quantities, when the goal is to generate a credible (as opposed to precise) simulation, geometric integrators allow one to take larger step sizes. Indeed, if the scheme is stable and geometric regardless of step size, the simulation will be credible even at large step sizes, and will not explode. One popular use of the Stormer-Verlet integrator is to simulate water surfaces in computer graphics using the graphics hardware.

By the way, the relationship between area under curve and location of particle at next step is essentially the relationship between area under curve and derivative, which is the fundamental theorem of calculus. Loisel 02:30, 17 Jun 2004 (UTC)

Thank you very much. The link to the Fundamental Theorem of Calculus was exactly what I was looking for (I had read the article on numerical integration, but that didn't help me make any connections). It's embarassing to be able to write code that implements these concepts to some degree, without understanding the concepts very well. I appreciate your responses as they help make more practical what was previously too theoretical for me to grasp.

The reader should know

Diberri removed the following sentence from the very beginning of the text:

It is recommended that the reader be familiar with algebra, derivatives, functions, and limits

I think this deserves more discussion. When I first saw this text I was also like "other pages of wikipedia don't have this kind of introduction!" but after some consideration I changed my opinion to be more like "perhaps its the other pages that should be changed". What do other people think about this?

IMO, it sounds odd for an encyclopedia to address the reader, which is why I moved these prerequisites to the related topics section. Wikipedia:WikiProject Mathematics is apparently in favor of these messages, though, which is why I didn't continue removing them from other math-related topics. --Diberri | Talk 23:28, Aug 25, 2004 (UTC)

I agree that It is recommended that the reader be familiar ... sounded strange, and I agree with its removal. For what it's worth, I'm not convinced that we need to take Wikipedia:WikiProject Mathematics all that seriously; my experience with proposal pages of various kinds in Wikipedia is that they attract the attention of too small a fraction of the editors to have a serious claim to authority. Wile E. Heresiarch 04:28, 19 Oct 2004 (UTC)

I am still hesitant on the question of this page, but on pages with more advanced mathematics such a warning is IMHO mandatory. I put something like that on most of my math pages. Gadykozma 11:34, 19 Oct 2004 (UTC)

The Integral of McShane

I was curious about this integral and searched in vain for an explanation. Maybe someone could write about it? I heard it is very general.

Mass?

The introduction mentions mass. How is it relevant? Brian j d | Why restrict HTML? | 10:25, 2005 May 8 (UTC)

It's the definite integral of density. Revolver 19:44, 2 September 2005 (UTC)

Integral of exp(-x²)

This is not a "nice function", according to the article. I think it is correct to say that it is not an elementary function - if somebody knows this, can they fix the article? Brian j d | Why restrict HTML? | 10:30, 2005 May 8 (UTC)

See Error function. Charles Matthews 16:29, 8 May 2005 (UTC)

Q:Why are integrals more difficult than derivatives?

This article mentions that integrals are often very hard to compute. In my calculus courses, it's been said many times that integrals are harder to compute than derivatives. Why is this? If they're inverse operations, wouldn't you intuitively expect them to be equally easy or difficult? What properties of the integral makes computing it more difficult than the derivative? -- Creidieki 18:13, 14 August 2005 (UTC)

There are lots of standard formulas for derivatives that you can put together and combine using other rules to compute the derivative for most any ordinary function you might run across. But if you have to find an antiderivative (which seems to be the subject of the article, which seems wrong to me), there are a lot less techniques. There is no "product formula" or "chain rule". The best you can do is reverse these to get integration by parts or u-substituions. But beyond that, you're on your own pretty much, it's fairly ad-hoc. Think of factoring/multiplying integers. Those are inverse processes, yet multiplying integers together is trivial and factoring is extremely nontrivial. Revolver 19:48, 2 September 2005 (UTC)

Dont say that before you seen the hard integrals. For example if you want to find the derivatives of cos^6 x it takes around 5 seconds, but for the integral of that you have to use de moivre's theorem and all to find it. It takes 5 minutes.

For cos^6 (x) can't u just use reduction formula? Thats a pain too, but whatever.

Merging with antiderivative?

See talk:antiderivative. Oleg Alexandrov (talk) 19:31, 4 November 2005 (UTC)

General case of integrals in terms of areas

I moved

For a function that takes positive and negative values, the integral is the sum of the areas of the parts above the x-axis minus the sum of those below.

here. This is just saying it is an algebraic sum (signed math). Also it could be taken to be techniucally inaccurate: The area below the axis could be interpreted as having a negative value and if so it should be added rather than subtracted.

If there is consensus to place it in the article again I will understand, however, it might still belong in a differnt location in the article. RJFJR 14:43, 7 December 2005 (UTC)

Area according to the normal definition is never negative.

I know the mentioned property can be derived from the property for non-negative functions, yet I think it is an odd omission not to mention it. If it bothers you to mention both we can mention only this general case, it includes the case of a non-negative function.--Patrick 00:00, 8 December 2005 (UTC)

Vote for new external link

Here is my site with integral example problems. Someone please put this link in the external links section if you think it's helpful and relevant. Tbsmith

http://www.exampleproblems.com/wiki/index.php/Calculus cool site ,Yet it's data could be merged with wikibooks or wikitext ,rather than here.---Procrastinator talk2me 00:08, 15 January 2006 (UTC)

Correct defenition of integral at need

The statement in the beginging of this article is confusing at best ,and completely non encyclopedic. Judging an integral by it's uses,is an arbitrary and incorect defenition. This article should begin with something like this:

An integral is a mathematical generalization of the concept of sum. It is used to calculate the total of all additive elements ,while the elements them selves are infinitesimally small,yet their rate of change is known by a function.(Ie: a temperature integral can produce the total energy in a given space)

Is it frequently used in calculating non countable quantities such as Mass(that has no discrete smallest value), Area, Volume, average location etc'.

This article is about the integral of a function of a single real variable.(for integrals of more than one variabeles ,see Multiple integral, or Vector analysis)

This should replace at least the first paragraph of this important article. Objections/corrections? :-)-Procrastinator talk2me 14:49, 15 January 2006 (UTC)

I don't like this wording. I find the current introduction to be better. It says:

In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. The process of finding integrals is integration, in its mathematical meaning. Unlike the closely-related process of differentiation, there are several possible definitions of integration, with different technical underpinnings. They are, however, compatible; any two different ways of integrating a function will give the same result when they are both defined.

Oleg Alexandrov (talk) 20:58, 15 January 2006 (UTC)

He's right though ,I was searching the net for a clear defenitoion ,and this is not it really. they can be used for various things ,but that doesn't define their actual qualities. although ,I must admint usually in wikipedia there are really good clear defenitions...for math things,and I learned a lot from here..

Please be more precise ,like the defenition it self you dont give a good description of what you dont like. You are more than welcome to change the wording ,or address my criticism at hand. You must concur that integral are Not merely about mass or volume.--Procrastinator@^talk2me 20:13, 16 January 2006 (UTC)

Now, is it you talking to yourself in the two paragraphs above?

I perfectly agree that the integral is not only about mass and volume. These are particular cases. That's why the article says that "integral is a generalization of the concepts of area and volume". Area and volume are just particular cases. It seems to me that you are trying to say the same thing, but the current article wording succeeds better than your suggested wording. Oleg Alexandrov (talk) 23:53, 16 January 2006 (UTC)

notation?

the article says "However, modern theories of integration are built from different foundations, and the traditional symbols have become no more than notation."

this is clearly false!

when writing f(x)dx, one means the differntial 1-form f(x) * dx, and you integrate the differntial form, not the function itself!

I'm not sure that the notation has to be taken to indicate a differential form. The dx can mean a couple of different things. Anyway, I don't like the sentence. What does it mean to be "no more than notation"? Isn't everything in mathematics "no more than notation", at least from a formalist perspective? I changed it to something more wishy-washy, but perhaps it should just be removed. -lethe ^{talk +} 17:27, 1 June 2006 (UTC)

Integral vs antiderivative

My revert is motivated by the fact that it is already clear enough from the current wording that integral may refer to antiderivative, and not just in India, but in many places. Insisting so much that in India it is official while in other places may not be is just distracting, not very relevant, and innappropriate so early in the text (the intro can be used for better things). Oleg Alexandrov (talk) 15:44, 10 August 2006 (UTC)

You do not get my point. I named India because it's the only country I know where the terminology is used in government publications. The point is that someone coming to the article about integrals may be bewildered by meeting with something unexpected, and the present wording does not make that clear enough. What we want to do is direct that reader to antiderivative right at the beginning. Also, calling the alternate term 'abuse' is biased. Loom91 10:51, 11 August 2006 (UTC)

Current wording is clear enough. It says that "integral" may refer to "antiderivative". At that point the user should visit antiderivative and read on that concept. Oleg Alexandrov (talk) 18:08, 11 August 2006 (UTC)

New addition

I am very confused as to what the sentence

One can also consider an integral to be the 'total' of a quantity that varies continuously (takes different values at every instant of time) over an interval.

in this revision is supposed to mean.

First, what is time in this context?

Second, "varying continuously" is not the same as "taking different values".

Third, the integral is defined for functions which are discontinuous also.

I removed that statement from the article is incorrect, please make your case here before putting it back in the article. Oleg Alexandrov (talk) 16:12, 14 August 2006 (UTC)

The problem with the current intro is that it makes no attempt to give an idea about what Integrals are. Saying integrals are a generalisation of the concept of sum conveys no information. The lead should summarise the main points of the article, and what an integral is is definitely a main point. Loom91 16:58, 15 August 2006 (UTC)

An integral is a sum. It is what you get by adding up lots of little pieces. An integral is not an area. Finding areas under curves is just one of the many applications of integration. I know you are taught in school that an integral is the area under a curve, but thats only part of the story. I have changed it back from area to sum. See the discussion above by the person whose spelling is poor but mathematical understanding is good. Paul Matthews 15:49, 16 August 2006 (UTC)

I never claimed that an integral is area under the curve, though that's a good application to build intuition and explain the Riemann definition. What the lead needs is something like what you say: "An integral is a sum. It is what you get by adding up lots of little pieces." The lead itself should contain some idea about what an integral is, and saying that it is an extension of the concept of sum does not do that. Loom91 14:00, 17 August 2006 (UTC)

Sorry, bad editing on my part. I wasn't objecting to you, Loom, but someone had changed the first sentence to say an integral was an area. Yes, I think the article needs more explanation and examples of what an integral is physically in the real world, with pictures. If I had a spare day I'd do it... Paul Matthews 14:40, 17 August 2006 (UTC)

I think Loom91's recently added remark that an integral gives a "measure of totality [...]" is spot-on, but the surrounding wording is far too clumsy. Fredrik Johansson 23:05, 17 August 2006 (UTC)

Can anyone tell my how to solve this $\int _{2}^{\infty }{\frac {1}{x^{2}-1}}\,dx$ Wendten20:02, 25. September 2006 (UTC)

You can make the next variable change: x=sin(y) Epicus15:27, 10. December 2006 (UTC)

That won't help. What you want to do is factor the denominator, and express

1/(x^{2}-1)

as

(1/2)/(x-1)-(1/2)/(x+1)

, then integrate. -- Doctormatt 01:31, 11 December 2006 (UTC)

Ive found a way to solve $\int x^{n}e^{-ax}dx$ and $\int x^{n}e^{ax}dx$ in e general way that is exact. is it ok if i add it to the list? if so where? —The preceding unsigned comment was added by 81.230.52.58 (talk • contribs) .

You are looking for the List of integrals of exponential functions article. Loisel 21:18, 18 October 2006 (UTC)

Wording

I'd encourage folks to move wording and nuance changes to this subsection. -- Near the bottom of the third paragraph of the article, describing the Riemann definition:

"A method for calculating this area using the concept of limit by dividing the area into successively thinner rectangular strips and taking the sum of their areas"

I decided to go ahead and change "successively thinner rectangular strips" to "arbitrarily thin rectangular strips" because I believe it to be more conceptually accurate in that there will be error in the integration if the strips have any thickness at all - however, perhaps it is less readable. Also conceptually accurate would be "infinitely thin rectangular strips". What wording is best? Change to infinitely thin if it's better.

Of course wording is signifigant in a major or even minor article. Fulvius 09:03, 28 February 2007 (UTC)

I absolutely hate "infinitely thin." I have no idea what it means for something to be "infinitely thin." I'm not entirely sure that makes sense. I prefer "arbitrarily thin," as that is unambiguous. –King Bee ^{(T • C)} 13:32, 28 February 2007 (UTC)

Good Article (GA) review

This is a really nice article, and would make a terrific GA. However, I think it doesn't quite pass all of the criteria yet. For this reason, I am putting it "On hold" for 7 days.

GA review (see here for criteria)

It is reasonably well written.
a (prose): b (MoS):
It is factually accurate and verifiable.
a (references): b (citations to reliable sources): c (OR):
It is broad in its coverage.
a (major aspects): b (focused):
It follows the neutral point of view policy.
a (fair representation): b (all significant views):
It is stable.
It contains images, where possible, to illustrate the topic.
a (tagged and captioned): b lack of images (does not in itself exclude GA): c (non-free images have fair use rationales):
Overall:
a Pass/Fail:

Since this is my first time reviewing an article, please let me explain myself below:

Although I noticed that "we" was used at one point in the article, I think it fits the exception for mathematical articles outlined at WP:MOS. Because of this, I passed criterion 1c.
This article is in need of inline citations and more complete references throughout, but especially in the "Symbolic integration" section and when citing historical information. This is, by far, the largest issue in my opinion preventing this from being a GA. Because 2a and 2b do not pass, I cannot make any judgment on 2c and 2d.
While there is a correctly tagged image, I would just like to point out that the image used has been superseded by a vector-based version. As a result, I marked 6a as neutral instead of passing it outright. On a sidenote, I think that the addition of another image or two (possibly one from Riemann integral or Rectangle method) would really add a lot to an already nice article.

I have placed the article “On hold” for the changes to be implemented, and I hope this helps. Hotstreets 00:32, 12 October 2006 (UTC)

I am failing the GA nomination since the concerns mentioned by Hotstreets have not been addressed in the last 10 days. Though much of the content of the article could be considered general knowledge, the referencing is simply not up to the GA standard. No other complaints though and I am listing it as an unreferenced GA. Eluchil404 17:20, 23 October 2006 (UTC)

Lebesgue integrates a wider class of functions?

I was under the impression that the set of functions that are Riemann integrable and the set of functions that are Lebesgue integrable are not comparable; i.e., there are functions that are Riemann but not Lebesgue, and vice versa. So, does the Lebesgue integral really integrate a "wider class" of functions? What does this mean? Also, I feel that an unfair portion of the article is dedicated to the Riemann integral, while I feel that the Lebesgue integral is at least as important. Should we change the focus a bit, or have a different section explaining Lebesgue integration? --King Bee 22:39, 4 December 2006 (UTC)