Talk:Integral/Archive 1

353:" 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. 677:)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 190:

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.

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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,

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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.

885:(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. 31: 1898: 1888: 2027: 1993: 1977: 1967: 1943: 1933: 1861: 1847: 170:

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?

2061: 2041: 2017: 1912: 258:-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, 374:

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

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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

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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

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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

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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

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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

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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

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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

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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

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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,

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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

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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

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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

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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

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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.

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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.)

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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

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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,

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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

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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

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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

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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

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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.

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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? --

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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.

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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

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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.

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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.

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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

1123:, 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. 1011:

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:

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When someone types in "integration" or "integrate", they get Knowledge rules for page integration for integrate and Mathematics Integral for Integration. In technology, integrate means "to integrate to application or purpose" :

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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

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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

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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,

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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

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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. –

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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,

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all that seriously; my experience with proposal pages of various kinds in Knowledge is that they attract the attention of too small a fraction of the editors to have a serious claim to authority.

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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.

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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.

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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

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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.

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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

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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?

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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.

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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:

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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

1777:"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" 107:

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.

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

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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.

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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.

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In computer graphics, when learning about physical simulations (e.g. particle systems, physically based animation), I see mention over and over again of

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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:

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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.

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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.

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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.

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have changed it back from area to sum. See the discussion above by the person whose spelling is poor but mathematical understanding is good.

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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.

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conceptually accurate would be "infinitely thin rectangular strips". What wording is best? Change to infinitely thin if it's better.

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Although I noticed that "we" was used at one point in the article, I think it fits the exception for mathematical articles outlined at

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IMO, it sounds odd for an encyclopedia to address the reader, which is why I moved these prerequisites to the related topics section.

593:. Some people may have had a lousy education and look towards this article for guidance. It would be swell if someone added that. 768:, but on pages with more advanced mathematics such a warning is IMHO mandatory. I put something like that on most of my math pages. 751: 728: 1219:

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

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the information is invisible to the interpretive lens and not available for extraction. Integration brings a whole world to light.

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small,yet their rate of change is known by a function.(Ie: a temperature integral can produce the total energy in a given space)

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is apparently in favor of these messages, though, which is why I didn't continue removing them from other math-related topics. --

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I'd really appreciate it if someone could add a paragraph or two clarifying the relevance of integration to physics simulations

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Current wording is clear enough. It says that "integral" may refer to "antiderivative". At that point the user should visit

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I removed that statement from the article is incorrect, please make your case here before putting it back in the article.

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contain some idea about what an integral is, and saying that it is an extension of the concept of sum does not do that.

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

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For a function that takes positive and negative values, the integral is the sum of the areas of the parts above the

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This should replace at least the first paragraph of this important article. Objections/corrections? :-)-

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sounded strange, and I agree with its removal. For what it's worth, I'm not convinced that we need to take

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college. Kudos to mathematicians. They are definitely a step higher in the IQ scale than the rest of us.

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

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which I have shared ideas similar though less developed. It is a concept I have never seen in a

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I have placed the article “On hold” for the changes to be implemented, and I hope this helps.

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

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cool site ,Yet it's data could be merged with wikibooks or wikitext ,rather than here.---

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Is it frequently used in calculating non countable quantities such as Mass(that has no

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

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properties of the integral makes computing it more difficult than the derivative? --

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they help make more practical what was previously too theoretical for me to grasp.

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

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

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If you wish to start a new discussion or revive an old one, please do so on the

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in e general way that is exact. is it ok if i add it to the list? if so where?

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innappropriate so early in the text (the intro can be used for better things).

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

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

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but the current article wording succeeds better than your suggested wording.

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The symbols {,} and epsilon are used without definition or even reference to

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

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right at the beginning. Also, calling the alternate term 'abuse' is biased.

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

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mathematics are a bit more philosophical, exempli gratia, Isaac Newton.

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Of course wording is signifigant in a major or even minor article.

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I am failing the GA nomination since the concerns mentioned by

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

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An integral is a mathematical generalization of the concept of

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solvers (they are called numerical integrators.) Euler is u

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clear enough. What we want to do is direct that reader to

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Area according to the normal definition is never negative.

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the tool mathematics to implement, whereas those who

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Q:Why are integrals more difficult than derivatives?

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The introduction mentions mass. How is it relevant?

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No mention of integration pertaining to simulation?

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Because of this, I passed criterion 1c. 1709: 1699: 1690: 1657: 1647: 1638: 1591: 1580: 1551: 1540: 1532: 1502: 1490: 1485: 1433: 1423: 1417: 1412: 1406: 537:{\displaystyle \int _{u_{0}}^{u}f(x)\,dx} 508: 501: 496: 490: 463: 456: 451: 425: 406: 393: 313: 150:Well, here are some examples. I used the 1041:This article is about the integral of a 1614:{\displaystyle (1/2)/(x-1)-(1/2)/(x+1)} 1448: 764:I am still hesitant on the question of 526: 482:instead of the traditonal and accepted 1326:First, what is time in this context? 1115:. The process of finding integrals is 1007:Correct defenition of integral at need 44:Do not edit the contents of this page. 1401:Can anyone tell my how to solve this 699:Thank you very much. The link to the 7: 679:with a fixed, non-adaptive step size 152:Mathematica integrator web interface 1678:{\displaystyle \int x^{n}e^{-ax}dx} 944:-axis minus the sum of those below. 556:Ah, today I finanlly stumbled upon 1911: 1727:{\displaystyle \int x^{n}e^{ax}dx} 1418: 24: 752:Knowledge:WikiProject Mathematics 729:Knowledge:WikiProject Mathematics 472:{\displaystyle \left_{u_{0}}^{u}} 327:{\displaystyle \lfloor x\rfloor } 2059: 2039: 2025: 2015: 1991: 1975: 1965: 1941: 1931: 1910: 1896: 1886: 1859: 1845: 1469:20:02, 25. September 2006 (UTC) 29: 2081:make any judgment on 2c and 2d. 2060: 2040: 2016: 1893:(citations to reliable sources) 1477:15:27, 10. December 2006 (UTC) 1223:I'm not sure that the notation 701:Fundamental Theorem of Calculus 689:fundamental theorem of calculus 585:Use of symbols from set theory? 1608: 1596: 1588: 1574: 1568: 1556: 1548: 1534: 1514: 1495: 1154:merely about mass or volume.-- 813:It's the definite integral of 663:ordinary differential equation 634:computer animation and games. 614:as a key step in the process. 523: 517: 443: 437: 1: 1804:13:32, 28 February 2007 (UTC) 1789:09:03, 28 February 2007 (UTC) 1626:01:31, 11 December 2006 (UTC) 890:19:48, 2 September 2005 (UTC) 822:19:44, 2 September 2005 (UTC) 2137:22:39, 4 December 2006 (UTC) 2121:17:20, 23 October 2006 (UTC) 2102:00:32, 12 October 2006 (UTC) 2026: 1992: 1976: 1966: 1942: 1932: 1897: 1887: 1860: 1846: 1765:21:18, 18 October 2006 (UTC) 1199:23:53, 16 January 2006 (UTC) 1163:20:13, 16 January 2006 (UTC) 1137:20:58, 15 January 2006 (UTC) 1074:14:49, 15 January 2006 (UTC) 1002:00:08, 15 January 2006 (UTC) 973:00:00, 8 December 2005 (UTC) 957:14:43, 7 December 2005 (UTC) 927:19:31, 4 November 2005 (UTC) 1520:{\displaystyle 1/(x^{2}-1)} 1396:23:05, 17 August 2006 (UTC) 1387:14:40, 17 August 2006 (UTC) 1377:14:00, 17 August 2006 (UTC) 1367:15:49, 16 August 2006 (UTC) 1356:16:58, 15 August 2006 (UTC) 1345:16:12, 14 August 2006 (UTC) 1299:18:08, 11 August 2006 (UTC) 1277:10:51, 11 August 2006 (UTC) 1262:15:44, 10 August 2006 (UTC) 875:18:13, 14 August 2005 (UTC) 2152: 1289:and read on that concept. 1243:Integral vs antiderivative 979:Vote for new external link 846:| 10:30, 2005 May 8 (UTC) 809:| 10:25, 2005 May 8 (UTC) 163:(3) x Exp (1/2) (1+Erf])^2 1633:Ive found a way to solve 1038:, average location etc'. 772:11:34, 19 Oct 2004 (UTC) 739:23:28, Aug 25, 2004 (UTC) 598:03:58, 6 March 2007 (UTC) 262:03:16, 11 Feb 2004 (UTC) 166:(4) Exp/(Exp+Exp) Exp Exp 1810:Good Article (GA) review 1756:You are looking for the 1237:17:27, 1 June 2006 (UTC) 758:04:28, 19 Oct 2004 (UTC) 695:02:30, 17 Jun 2004 (UTC) 656:21:53, 16 Jun 2004 (UTC) 571:17:09, 22 May 2004 (UTC) 551:04:35, 22 May 2004 (UTC) 338:07:52 25 Jun 2003 (UTC) 294:01:48, 23 May 2004 (UTC) 272:19:33, 21 May 2004 (UTC) 126:02:00, 23 May 2004 (UTC) 115:19:28, 21 May 2004 (UTC) 1972:(all significant views) 1835:reasonably well written 1216:this is clearly false! 1095:is a generalization of 859:16:29, 8 May 2005 (UTC) 780:The Integral of McShane 244:16:14, 2 Jun 2004 (UTC) 218:15:36, 2 Jun 2004 (UTC) 194:07:22, 2 Jun 2004 (UTC) 174:04:42, 2 Jun 2004 (UTC) 139:11:09, 1 Jun 2004 (UTC) 2032:(non-free images have 2012:(tagged and captioned) 1728: 1679: 1615: 1521: 1460: 1323:is supposed to mean. 708:The reader should know 568:Lindberg G Williams Jr 548:Lindberg G Williams Jr 538: 473: 328: 1962:(fair representation) 1953:neutral point of view 1921:broad in its coverage 1740:comment was added by 1729: 1680: 1621:, then integrate. -- 1616: 1522: 1461: 646:differential equation 642:Numerical integration 539: 474: 329: 92:I have removed this: 42:of past discussions. 1689: 1637: 1531: 1484: 1405: 489: 392: 312: 2034:fair use rationales 1422: 915:talk:antiderivative 828:Integral of exp(-x) 558:fractional calculus 513: 468: 304:User:David Martland 1870:factually accurate 1724: 1675: 1611: 1517: 1456: 1449: 1408: 844:Why restrict HTML? 807:Why restrict HTML? 756:Wile E. Heresiarch 650:Runge-Kutta method 576:Warning to Readers 534: 527: 492: 469: 395: 324: 291:Wile E. Heresiarch 260:Wile E. Heresiarch 242:Wile E. Heresiarch 216:Wile E. Heresiarch 172:Wile E. Heresiarch 160:(2) Exp Exp ^2]/x 157:(1) Exp BesselK ] 123:Wile E. Heresiarch 1753: 1446: 1393:Fredrik Johansson 1053:Multiple integral 1030:smallest value), 419: 286:Lebesgue integral 280:I agree that the 85: 84: 54: 53: 48:current talk page 2143: 2091:Rectangle method 2087:Riemann integral 2063: 2062: 2043: 2042: 2029: 2028: 2019: 2018: 1995: 1994: 1979: 1978: 1969: 1968: 1945: 1944: 1935: 1934: 1914: 1913: 1900: 1899: 1890: 1889: 1863: 1862: 1849: 1848: 1735: 1733: 1731: 1730: 1725: 1717: 1716: 1704: 1703: 1684: 1682: 1681: 1676: 1668: 1667: 1652: 1651: 1620: 1618: 1617: 1612: 1595: 1584: 1555: 1544: 1526: 1524: 1523: 1518: 1507: 1506: 1494: 1465: 1463: 1462: 1457: 1447: 1445: 1438: 1437: 1424: 1421: 1416: 1158: 1143:lot from here.. 1071: 999: 856:Charles Matthews 543: 541: 540: 535: 512: 507: 506: 505: 478: 476: 475: 470: 467: 462: 461: 460: 450: 446: 433: 432: 424: 420: 418: 407: 333: 331: 330: 325: 88:Initial comments 63: 56: 55: 33: 32: 26: 2151: 2150: 2146: 2145: 2144: 2142: 2141: 2140: 2129: 2114:unreferenced GA 1950:It follows the 1928:(major aspects) 1812: 1772: 1736:—The preceding 1705: 1695: 1687: 1686: 1653: 1643: 1635: 1634: 1529: 1528: 1498: 1482: 1481: 1429: 1428: 1403: 1402: 1337:Oleg Alexandrov 1309: 1291:Oleg Alexandrov 1254:Oleg Alexandrov 1245: 1211: 1191:Oleg Alexandrov 1157:Procrastinator@ 1156: 1129:Oleg Alexandrov 1121:differentiation 1069: 1057:Vector analysis 1021:infinitesimally 1009: 997: 981: 934: 919:Oleg Alexandrov 911: 867: 830: 793: 782: 710: 676: 672: 668: 608: 587: 578: 497: 487: 486: 452: 411: 402: 401: 400: 396: 390: 389: 310: 309: 308:Something like 90: 59: 30: 22: 21: 20: 12: 11: 5: 2149: 2147: 2128: 2125: 2124: 2123: 2095: 2094: 2082: 2078: 2067: 2066: 2065: 2064: 2046: 2045: 2044: 1998: 1997: 1996: 1982: 1981: 1980: 1948: 1947: 1946: 1917: 1916: 1915: 1866: 1865: 1864: 1830: 1829: 1811: 1808: 1807: 1806: 1792: 1771: 1768: 1723: 1720: 1715: 1712: 1708: 1702: 1698: 1694: 1674: 1671: 1666: 1663: 1660: 1656: 1650: 1646: 1642: 1631: 1630: 1629: 1628: 1610: 1607: 1604: 1601: 1598: 1594: 1590: 1587: 1583: 1579: 1576: 1573: 1570: 1567: 1564: 1561: 1558: 1554: 1550: 1547: 1543: 1539: 1536: 1516: 1513: 1510: 1505: 1501: 1497: 1493: 1489: 1455: 1452: 1444: 1441: 1436: 1432: 1427: 1420: 1415: 1411: 1400: 1359: 1358: 1317: 1316: 1308: 1305: 1304: 1303: 1302: 1301: 1287:antiderivative 1280: 1279: 1270:antiderivative 1244: 1241: 1240: 1239: 1210: 1207: 1206: 1205: 1204: 1203: 1202: 1201: 1181: 1180: 1179: 1178: 1177: 1176: 1168: 1167: 1166: 1165: 1140: 1139: 1126: 1125: 1124: 1077: 1076: 1066:Procrastinator 1008: 1005: 994:Procrastinator 980: 977: 976: 975: 964: 963: 933: 930: 910: 907:antiderivative 903: 902: 901: 893: 892: 883:antiderivative 866: 863: 862: 861: 852:Error function 829: 826: 825: 824: 792: 786: 781: 778: 777: 775: 762: 761: 760: 759: 741: 740: 720: 719: 709: 706: 697: 696: 684: 683: 674: 670: 666: 658: 657: 607: 604: 602: 586: 583: 577: 574: 554: 545: 544: 533: 530: 525: 522: 519: 516: 511: 504: 500: 495: 480: 479: 466: 459: 455: 449: 445: 442: 439: 436: 431: 428: 423: 417: 414: 410: 405: 399: 366: 346: 323: 320: 317: 300: 299: 298: 297: 296: 275: 274: 255: 254: 253: 252: 251: 250: 249: 248: 247: 246: 245: 228: 227: 226: 225: 224: 223: 222: 221: 220: 219: 202: 201: 200: 199: 198: 197: 196: 195: 180: 179: 178: 177: 176: 175: 167: 164: 161: 158: 155: 143: 142: 141: 140: 129: 128: 89: 86: 83: 82: 77: 74: 69: 64: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 2148: 2139: 2138: 2135: 2126: 2122: 2119: 2115: 2110: 2106: 2105: 2104: 2103: 2100: 2092: 2088: 2083: 2079: 2076: 2072: 2071: 2070: 2057: 2053: 2052: 2050: 2047: 2037: 2035: 2023: 2013: 2009: 2008: 2006: 2005: 1999: 1990: 1989: 1987: 1983: 1973: 1963: 1959: 1958: 1956: 1954: 1949: 1939: 1929: 1925: 1924: 1922: 1918: 1908: 1906: 1894: 1884: 1880: 1879: 1877: 1876: 1871: 1867: 1857: 1855: 1843: 1839: 1838: 1836: 1832: 1831: 1828:for criteria) 1827: 1823: 1821: 1817: 1816: 1815: 1809: 1805: 1802: 1801: 1800: 1794: 1793: 1791: 1790: 1787: 1782: 1778: 1775: 1769: 1767: 1766: 1763: 1759: 1754: 1751: 1747: 1743: 1739: 1721: 1718: 1713: 1710: 1706: 1700: 1696: 1692: 1672: 1669: 1664: 1661: 1658: 1654: 1648: 1644: 1640: 1627: 1624: 1605: 1602: 1599: 1592: 1585: 1581: 1577: 1571: 1565: 1562: 1559: 1552: 1545: 1541: 1537: 1511: 1508: 1503: 1499: 1491: 1487: 1479: 1478: 1476: 1472: 1471: 1470: 1468: 1453: 1450: 1442: 1439: 1434: 1430: 1425: 1413: 1409: 1398: 1397: 1394: 1389: 1388: 1385: 1384:Paul Matthews 1379: 1378: 1375: 1369: 1368: 1365: 1364:Paul Matthews 1357: 1354: 1349: 1348: 1347: 1346: 1342: 1338: 1333: 1330: 1327: 1324: 1322: 1314: 1313: 1312: 1306: 1300: 1296: 1292: 1288: 1284: 1283: 1282: 1281: 1278: 1275: 1271: 1266: 1265: 1264: 1263: 1259: 1255: 1250: 1242: 1238: 1235: 1230: 1226: 1222: 1221: 1220: 1217: 1214: 1208: 1200: 1196: 1192: 1187: 1186: 1185: 1184: 1183: 1182: 1174: 1173: 1172: 1171: 1170: 1169: 1164: 1161: 1159: 1153: 1148: 1147: 1146: 1145: 1144: 1138: 1134: 1130: 1127: 1122: 1118: 1114: 1110: 1106: 1102: 1098: 1094: 1090: 1086: 1082: 1081: 1079: 1078: 1075: 1072: 1067: 1063: 1062: 1061: 1060: 1058: 1054: 1048: 1044: 1039: 1037: 1033: 1029: 1024: 1022: 1018: 1013: 1006: 1004: 1003: 1000: 995: 991: 987: 986: 978: 974: 971: 966: 965: 961: 960: 959: 958: 955: 950: 946: 945: 941: 937: 931: 929: 928: 924: 920: 916: 908: 905:Merging with 904: 899: 898: 897: 891: 888: 884: 879: 878: 877: 876: 873: 864: 860: 857: 853: 849: 848: 847: 845: 841: 838: 835: 827: 823: 820: 816: 812: 811: 810: 808: 804: 801: 798: 790: 787: 785: 779: 776: 773: 771: 767: 757: 753: 749: 746:I agree that 745: 744: 743: 742: 738: 734: 730: 726: 725: 724: 718: 715: 714: 713: 707: 705: 702: 694: 690: 686: 685: 680: 664: 660: 659: 655: 651: 647: 643: 640: 639: 638: 635: 633: 628: 627:in some way? 625: 621: 617: 613: 605: 603: 600: 599: 596: 592: 584: 582: 575: 573: 572: 569: 565: 563: 559: 553: 552: 549: 531: 528: 520: 514: 509: 502: 498: 493: 485: 484: 483: 464: 457: 453: 447: 440: 434: 429: 426: 421: 415: 412: 408: 403: 397: 388: 387: 386: 383: 381: 377: 372: 365: 361: 357: 354: 349: 348:New Subject: 345: 344: 341:Or ⌊ x ⌋ ? - 339: 337: 318: 306: 305: 295: 292: 287: 283: 279: 278: 277: 276: 273: 270: 265: 264: 263: 261: 243: 238: 237: 236: 235: 234: 233: 232: 231: 230: 229: 217: 212: 211: 210: 209: 208: 207: 206: 205: 204: 203: 193: 188: 187: 186: 185: 184: 183: 182: 181: 173: 168: 165: 162: 159: 156: 153: 149: 148: 147: 146: 145: 144: 138: 133: 132: 131: 130: 127: 124: 119: 118: 117: 116: 113: 108: 105: 102: 98: 97: 93: 87: 81: 78: 75: 73: 70: 68: 65: 62: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 18:Talk:Integral 2130: 2096: 2068: 2055: 2048: 2031: 2021: 2011: 2001: 1985: 1971: 1961: 1951: 1937: 1927: 1920: 1902: 1892: 1883:(references) 1882: 1873: 1869: 1851: 1841: 1834: 1818: 1813: 1797: 1796: 1783: 1779: 1776: 1773: 1755: 1742:81.230.52.58 1632: 1399: 1390: 1380: 1370: 1360: 1334: 1331: 1328: 1325: 1318: 1310: 1307:New addition 1246: 1228: 1224: 1218: 1215: 1212: 1151: 1141: 1116: 1088: 1050: 1045:of a single 1040: 1025: 1014: 1010: 988: 982: 951: 947: 943: 939: 938: 935: 912: 894: 868: 831: 794: 783: 774: 765: 763: 747: 721: 716: 711: 700: 698: 678: 636: 631: 629: 623: 619: 615: 611: 609: 601: 588: 579: 566: 555: 546: 481: 384: 379: 375: 370: 367: 362: 358: 355: 350: 347: 340: 307: 301: 256: 109: 106: 103: 99: 95: 94: 91: 60: 43: 37: 1117:integration 968:function.-- 620:Runge-Kutta 612:integration 36:This is an 2118:Eluchil404 2109:Hotstreets 2099:Hotstreets 1875:verifiable 1623:Doctormatt 1232:removed. - 1049:variable.( 591:set theory 2056:Pass/Fail 2002:contains 1938:(focused) 1760:article. 1209:notation? 872:Creidieki 770:Gadykozma 766:this page 562:text book 343:The Anome 336:The Anome 80:Archive 5 72:Archive 3 67:Archive 2 61:Archive 1 2134:King Bee 1799:King Bee 1750:contribs 1738:unsigned 1321:revision 1319:in this 1093:function 1089:integral 1085:calculus 1043:function 1028:discrete 936:I moved 887:Revolver 819:Revolver 637:Thanks! 282:integral 2049:Overall 1842:(prose) 1786:Fulvius 1770:Wording 1467:Wendten 1070:talk2me 998:talk2me 985:Tbsmith 970:Patrick 815:density 733:Diberri 39:archive 2075:WP:MOS 2004:images 1986:stable 1984:It is 1955:policy 1919:It is 1868:It is 1833:It is 1822:review 1762:Loisel 1475:Epicus 1382:it... 1374:Loom91 1353:Loom91 1274:Loom91 1249:revert 1111:, and 1105:volume 1087:, the 1036:Volume 723:this? 693:Loisel 624:Verlet 380:define 269:Loisel 192:Loisel 137:Loisel 112:Loisel 1824:(see 1234:lethe 1113:total 1091:of a 1055:, or 954:RJFJR 834:Brian 797:Brian 673:+u'(x 654:DrBob 616:Euler 371:range 334:? -- 16:< 1872:and 1826:here 1746:talk 1685:and 1341:talk 1295:talk 1258:talk 1195:talk 1133:talk 1101:mass 1097:area 1047:real 1032:Area 923:talk 913:See 850:See 789:Mass 737:Talk 632:a la 622:and 595:Mbac 2116:. 2089:or 2000:It 1854:MoS 1752:) . 1527:as 1247:My 1225:has 1152:Not 1109:sum 1083:In 1017:sum 667:k+1 376:use 352:--> 2058:: 2054:a 2051:: 2038:: 2030:c 2024:: 2020:b 2014:: 2010:a 1988:. 1974:: 1970:b 1964:: 1960:a 1957:. 1940:: 1936:b 1930:: 1926:a 1923:. 1909:: 1905:OR 1901:c 1895:: 1891:b 1885:: 1881:a 1878:. 1858:: 1850:b 1844:: 1840:a 1837:. 1820:GA 1748:• 1693:∫ 1659:− 1641:∫ 1572:− 1563:− 1509:− 1440:− 1419:∞ 1410:∫ 1343:) 1297:) 1260:) 1229:dx 1197:) 1135:) 1107:, 1103:, 1099:, 1034:, 925:) 917:. 854:. 842:| 817:. 805:| 735:| 691:. 669:=u 618:, 494:∫ 427:− 322:⌋ 316:⌊ 76:→ 2036:) 1907:) 1903:( 1856:) 1852:( 1744:( 1722:x 1719:d 1714:x 1711:a 1707:e 1701:n 1697:x 1673:x 1670:d 1665:x 1662:a 1655:e 1649:n 1645:x 1609:) 1606:1 1603:+ 1600:x 1597:( 1593:/ 1589:) 1586:2 1582:/ 1578:1 1575:( 1569:) 1566:1 1560:x 1557:( 1553:/ 1549:) 1546:2 1542:/ 1538:1 1535:( 1515:) 1512:1 1504:2 1500:x 1496:( 1492:/ 1488:1 1454:x 1451:d 1443:1 1435:2 1431:x 1426:1 1414:2 1339:( 1293:( 1256:( 1193:( 1131:( 1059:) 942:x 921:( 909:? 840:d 837:j 803:d 800:j 791:? 675:k 671:k 532:x 529:d 524:) 521:x 518:( 515:f 510:u 503:0 499:u 465:u 458:0 454:u 448:] 444:) 441:x 438:( 435:f 430:1 422:) 416:x 413:d 409:d 404:( 398:[ 319:x 50:.

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