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defined simply as the inverse of differentiation. Even
Lacroix wrote that “the integral calculus is the inverse of the differential calculus, its object being to ascend from the differential coefficients to the function from which they are derived.” Although Leibniz had developed his notation to remind one of the integral as an infinite sum of infinitesimal areas, the problems inherent in the use of infinities convinced eighteenth-century mathematicians to take the notion of the indefinite integral, or antiderivative, as their basic notion for the theory of integration. They of course recognized that one could evaluate areas not only by use of antiderivatives but also by various approximation techniques. But it was Cauchy who first took these techniques as fundamental and proceeded to construct a theory of definite integrals upon them.
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defined simply as the inverse of differentiation.… Although
Leibniz had developed his notation to remind one of the integral as an infinite sum of infinitesimal areas, the problems inherent in the use of infinities convinced eighteenth-century mathematicians to take the notion of the indefinite integral, or antiderivative, as their basic notion for the theory of integration. They of course recognized that one could evaluate areas not only by use of antiderivatives but also by various approximation techniques. But it was Cauchy who first took these techniques as fundamental and proceeded to construct a theory of definite integrals upon them.
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point, but there is evidence that Cauchy was reproached by the directors of the school. He was told that, because the École
Polytechnique was basically an engineering school, he should use class time to teach applications of differential equations rather than to deal with questions of rigor. Cauchy was forced to conform and announced that he would no longer give completely rigorous demonstrations. He evidently then felt that he could not publish his lectures on the material, because they did not reflect his own conception of how the subject should be handled.
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definition. Cauchy also proved general existence theorems in the theory of differential equations. Instead of asking how to integrate a special function or a special differential equation (that is, finding an analytic expression for the solution), Cauchy began the process of establishing the existence of the integral for a wide class of functions (or differential equations). He thus started an important process towards a qualitative mathematics which was carried further by Sturm-Liouville theory and by
Poincare (see Chapter 11).
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1213:. Moreover, having a rigorous formalization of integrals requires not only a formal definition of limits, but also the proof that the limit does not depend on the way of dividing the interval of integration. So, my interpretation of Katz's quotation is that "Cauchy was the first to define integrals from limits", but this does not imply that it is not Riemann who "first formalized rigorously integrals, using limits". So, unless better sources are provided, section
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2124:, Cauchy presented the details of a rigorous definition of the integral using sums. Cauchy probably took his definition from the work on approximations of definite integrals by Euler and by Lacroix. But rather than consider this method a way of approximating an area, presumably understood intuitively to exist, Cauchy made the approximation into a definition.
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2182:.… Riemann now asked a question that Cauchy had not: In what cases is a function integrable and in what cases not? Cauchy himself had only shown that a certain class of functions was integrable, but had not tried to find all such functions. Riemann, on the other hand, formulated a necessary and sufficient condition for a finite function
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There is a curious story connected with Cauchy’s treatment of differential equations. Cauchy never published an account of this second-year course, and it is only recently that proof sheets for the first thirteen lectures of the course have come to light. It is not clear why these notes stop at this
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On the other hand, I disagree with formulations such as "the first to have formalized...": In this context, the concept of formalization as well as the concept of rigor has evolved over the time (let us recal that the first formal definition of the real numbers dates from the second half of the 19th
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To me, it is clear that Cauchy is the first to rigorously define integrals in the modern sense, unless somebody else did before him. He did it at the latest in 1823, thirty years before
Riemann On the other hand, Cauchy seemed mainly interested in actually integrating functions, rather than studying
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What I'm taking out of this is that we can probably say that
Riemann can be credited with turning integrable functions into an object of study, and this is likely why so many people say that he's the first to rigorously define the integral. Incidentally, since Riemann's and Cauchy's definitions of
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Cauchy broke radically from his predecessors with his definition of the integral… Leibniz had considered integrals as sums of infinitesimals but from the
Bernoullis onwards it had been customary to define integration as the inverse process of differentiation. This made the indefinite integral the
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Euler and his contemporaries had already used left sums to approximate integrals, and
Lacroix and Poisson had tried to prove that they converge to the integral in a suitable sense. One can find many elements of Cauchy's arguments in these papers as well as in Lagrange's proof of the fundamental
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In spite of the appeal of
Lagrange’s method in England, Cauchy, back in France, found that this method was lacking in “rigor.” Cauchy in fact was not satisfied with what he believed were unfounded manipulations of algebraic expressions, especially infinitely long ones. Equations involving these
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In his calculus texts, Cauchy defined the integral as a limit of a sum rather than as an antiderivative, as had been common in the eighteenth century. His extension of this notion of the integral to the domain of complex numbers led him to begin the development of complex analysis by the 1820s.
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putting the focus on the collection of functions that are integrable according to some notion of integrability, rather than defining a notion of integrability only for the purpose of being able to prove certain desired integrability properties. Regarding (B), I believe this was the first time a
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would have been almost forgotten during the 18th century. On the other hand, it seems true that, during the 18th century, the standard method for computing an integral was to compute first the antiderivative. One must recall that Cauchy was not teaching to future academic people, but to future
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Cauchy’s treatment of the derivative, although using his new definition of limits, was closely related to the treatments in the works of Euler and
Lagrange. Cauchy’s treatment of the integral, on the other hand, broke entirely new ground. Recall that, in the eighteenth century, integration was
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Cauchy’s treatment of the derivative, although using his new definition of limits, was closely related to the treatments in the works of Euler and Lagrange. Cauchy’s treatment of the integral, on the other hand, broke entirely new ground. Recall that, in the eighteenth century, integration was
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Even though what it means to “rigorously formalize” something is somewhat subjective, I would argue that Cauchy “rigorously formalized” integration (of piecewise continuous functions) some decades before Riemann. Indeed, the same reference (Katz 2009, pp. 776–777) seems to say the same thing:
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does not converge to the function. Thus, because from 1813 he was teaching at the École Polytechnique, Cauchy began to rethink the basis of the calculus entirely. In 1821, at the urging of several of his colleagues, he published his Cours d’analyse de l’École Royale Polytechnique in which he
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In the case of the integral the earlier approximation techniques led Cauchy to a definition which allowed him to prove the existence of the integral for a specific type of functions. No one seems to have asked this existence question before, nor could it have been answered with the earlier
1502:, rather than “defining” the integral as an antiderivative. He was only interested in integrating functions with finitely many discontinuities, though, and in fact he mainly focuses on continuous functions. For such functions, he shows (implicitly using the fact that a continuous function
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Cauchy followed Fourier when he focused on the definite integral, but instead of relying on a vague notion of area, Cauchy defined the definite integral as the limit of a "left sum". This was much more precise and it allowed him to prove that the integral exists for a continuous
994:. This article presents one of these conventions, in the "Formal definition" section. I'm not an expert on the history, but I think that it's based on Riemann's original formulation. A different convention leads to the upper and lower Darboux integrals, which are a bit simpler.
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theorem of calculus (see (Grabiner 1981, Chapter 6)), and it is very possible that Cauchy built on these sources. Yet Cauchy's treatment is much clearer… nd most importantly, Cauchy changed the technique from being a numerical approximation procedure to being a definition....
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As Euler knew the concept of limits, it seems unbelievable that he did not know a definition of integrals in terms of limits. So, it seems difficult to decide who was the first to use limits for defining integration, and what is exactly the contribution of
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introduced new methods into the foundations of the calculus. We will study Cauchy’s ideas on limits, continuity, convergence, derivatives, and integrals in the context of an analysis of this text as well as its sequel of 1823,
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the integral yield the same set of functions, we should maybe say that functions are Cauchy-integrable (or Cauchy-Riemann integrable) rather than Riemann-integrable, although that ship has sailed more than a hundred years ago.
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It's not quite clear what condition (A) is, but it seems to be some precursor to the modern theorem that a bounded function is Riemann-integrable iff it is continuous almost everywhere (see the StackExchange response for more
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It seems pretty clear to me that Katz views Cauchy as having first rigorously formalized the integral, as part of a larger program of introducing rigor in analysis in general, and Riemann having expanded on Cauchy's work.
1979:, by analyzing these texts and coming to our own judgment based on our knowledge of the math. It would be safer if we had reliable secondary sources explicitly saying that so-and-so was the first to formalize integrals.
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In the century since Newton and Leibniz the fundamental theorem of the calculus had become regarded as allowing the integral of a function to be regarded as the opposite of its derivative; the integral of a function
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that discusses Riemann's contribution to the theory of integration. It is well written and has a lot of references and is well worth reading in full, but of particular interest for us is the following paragraph:
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In 1853, Georg Bernhard Riemann (1826–1866) attempted to generalize Dirichlet’s result by first determining precisely which functions were integrable according to Cauchy’s definition of the integral
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Section 22.1 is “Rigor in Analysis”, and subsections 22.1.1-5 (“Limits“, “Continuity”, “Convergence”, “Derivatives”, and “Integrals”) are essentially all about Cauchy's work. The section opens with:
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engineers. So, the fact that many antiderivatives of common functions cannot be written in closed form was certainly a strong motivation for emphasizing on integrals rather than on antiderivatives.
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I think what any of us finds believable or not should have less importance than what solid secondary sources are saying on the question. Katz is extremely clear that Cauchy is the first to have
2484:(putting the limits of integration at the top and bottom of the integral sign is in fact Fourier's idea) and stressed that it meant the area between the curve and the axis (Fourier 1822, §229).
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function that was continuous on a dense set and discontinuous on another dense set had been defined (or even contemplated, for that matter). A well known example of such a function is the
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which functions are integrable, or studying integrable functions as a class. As far as I'm aware, all he ever considers are piecewise continuous functions, which he shows to be integrable.
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Cauchy firmly reversed this trend, and restored an independent existence on the integral. In the second part of the Résumé of 1823 he defined the integral as a limit of sums of areas.
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expressions were only true for certain values, those values for which the infinite series was convergent. In particular, Cauchy discovered that the Taylor series for the function
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Besides, the first to provide a coherent theory of integration that includes the fundamental theorem of calculus as a corollary… is Cauchy, who proves the theorem in his
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primary concept and had made integral calculus an appendix to differential calculus. Fourier was the first to change this picture… he focused on the definite integral
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It is clear that Cauchy defined integrals as limits of sums of areas of small rectangles. But, I am not sure that he used a formal definition of limits. According to
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Rubi, a computer algebra system rule-based integrator, pattern matches an extensive system of symbolic integration rules to integrate a wide variety of integrands.
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In particular, it was Cauchy, not Riemann, who first used limits to define the integral of a function. Is there any reason not to change the text to reflect this?
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So I think that you're asking why the article presents Riemann integrals instead of Darboux integrals. That's a fair question. I don't know the answer.
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Thanks for raising this issue. It would help to have a clearer statement of the timeline, because Cauchy and Riemann overlapped in time. According to
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Once we establish the basic facts, then it would be good to understand why so many authors seem to attribute the first rigorous integral to Riemann.
1177:(not a reliable source, I know), Riemann presented the Riemann integral in 1854. When did Cauchy do his integral work? It's not explicitly said at
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It seems that Katz did a confusion between "definition" and "computation": If the above quotation would be taken literaly, this would mean that
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Rich, Albert; Scheibe, Patrick; Abbasi, Nasser (16 December 2018), "Rule-based integration: An extensive system of symbolic integration rules",
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century, and thus that Cayley did not have a formal definition of the real numbers used in its "formal" definition of integrals).
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at the time. There may be suggestions below for improving the article. Once these issues have been addressed, the article can be
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2856:(see e.g., Katz p. 778 or Lützen p. 171). Not sure about Fubini or Stokes' theorems, it's an interesting question for sure.
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the integral as the limit of a sum (see above quotes). Here's a few other secondary sources saying the same thing.
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includes an explicit discussion that the way of cutting intervals does not change the limit value of the integral.
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that is continuous in a given interval was defined as follows. He divided the interval into n equal subintervals
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I doubt practical matters of engineering was a strong driver of how he shaped his course prior to that point.
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using this condition to prove the integrability of a certain function having a dense set of discontinuities;
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Once we understand that, if everything holds up, then multiple Knowledge articles will need to be changed.
2845:(Gray, Jeremy. The real and the complex: a history of analysis in the 19th century. Cham: Springer, 2015.)
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Chapter 22 is titled “Analysis in the Nineteenth Century”. In the chapter introduction (p. 765), we read:
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here, but he's not actually specifying the codomain of the function, it might actually be more general.
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With respect to the infinitesimals, it's less clear, but the word doesn't seem to appear in the proof.
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Is there a reason this article doesn’t include the standard definition for the Riemann integral? i.e.
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While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of
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is a generalization of Ramanujan's master theorem that can be applied to a wide range of integrals.
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giving a necessary and sufficient condition for integrability based on the behavior of a function;
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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I read somewhere that Cauchy also argues that we obtain the same definite integral if we define
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was the first to provide a coherent theory of integration that includes as corollaries the
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Section 22.1.6 discusses Fourier's work, then in Section 22.1.7 (“The Riemann Integral”):
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exist to draw attention and ensure that more editors mediate or comment on the dispute.
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Riemann further developed and extended these ideas in the middle of the century.
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Cauchy seems to have been the first to define the integral of a function
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to be integrable: “If, with the infinite decrease of all the quantities
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Some images to illustrate the Informal discussion section. Like what?--
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I agree. I personally think we have everything we need in (Katz 2009).
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along with two brief sections on the two definitions of the integral.
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2504:(Lützen, Jesper. "The foundation of analysis in the 19th century."
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In the History section, the subsection Formalization begins with:
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the section on Computing integrals could do with some expansion.
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this limit value does not depend on the choice of partitions
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Gonzalez, Ivan; Jiu, Lin; Moll, Victor H. (1 January 2020),
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This is all great to read. Thanks for putting it together.
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always becomes infinitely small in the end, then the sum
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of the intervals in which the variations of the function
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treatment of integrals with regard to differential forms.
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For these reasons, I suggest to theplace the sentence
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The Cauchy integral of a function of a real variable
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to determine whether its use and function meets the
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of the decision if they believe there was a mistake.
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63:New to Knowledge? Welcome!
2968:
2120:In the second part of his
1270:Definition of the integral
816:03:12, 23 March 2023 (UTC)
691:bracket integration method
326:. Editors may also seek a
2729:, and considered the sum
628:21:49, 28 June 2007 (UTC)
506:
439:
418:
340:
336:
93:Be welcoming to newcomers
22:Skip to table of contents
2430:Here's Lützen (p. 170):
1037:redirects for discussion
1019:Redirects for discussion
513:project's priority scale
21:
2882:21:25, 5 May 2024 (UTC)
2577:with the property that
2416:13:16, 5 May 2024 (UTC)
2356:17:42, 4 May 2024 (UTC)
2333:{\displaystyle \delta }
2320:converges when all the
2293:{\displaystyle \sigma }
2224:{\displaystyle \delta }
1989:15:53, 4 May 2024 (UTC)
1757:15:43, 4 May 2024 (UTC)
1624:{\displaystyle \sigma }
1600:{\displaystyle \sigma }
1260:20:54, 3 May 2024 (UTC)
1227:16:42, 3 May 2024 (UTC)
1201:16:09, 3 May 2024 (UTC)
1168:14:39, 3 May 2024 (UTC)
470:WikiProject Mathematics
2912:B-Class vital articles
2836:
2816:
2723:
2671:
2670:{\displaystyle y=f(x)}
2630:
2571:
2542:
2478:
2334:
2314:
2294:
2274:
2245:
2225:
2205:
2176:
2092:
1940:
1875:
1836:
1792:
1733:
1695:Riemann's contribution
1679:
1625:
1601:
1581:
1542:
1496:
1437:
1314:
1110:
1100:
976:
935:
905:
749:10.1515/math-2020-0062
88:avoid personal attacks
2837:
2817:
2724:
2672:
2631:
2572:
2543:
2515:Here's Gray (p. 55):
2506:A history of analysis
2479:
2335:
2315:
2295:
2275:
2246:
2226:
2206:
2177:
2093:
1941:
1849:
1837:
1793:
1708:
1680:
1626:
1602:
1582:
1543:
1497:
1411:
1315:
1179:Augustin-Louis Cauchy
1105:
1076:
977:
975:{\displaystyle x_{i}}
936:
885:
397:level-4 vital article
320:good article criteria
228:Auto-archiving period
113:Neutral point of view
2826:
2733:
2681:
2646:
2581:
2570:{\displaystyle F(x)}
2552:
2541:{\displaystyle f(x)}
2523:
2438:
2324:
2304:
2284:
2273:{\displaystyle f(x)}
2255:
2235:
2215:
2204:{\displaystyle f(x)}
2186:
2136:
2013:
1846:
1811:
1780:
1639:
1615:
1591:
1556:
1506:
1324:
1278:
959:
830:
493:mathematics articles
362:Good article nominee
118:No original research
2842:tends to infinity.
2455:
2153:
1656:
1607:tends to zero, and
1320:using the quantity
1041:redirect guidelines
1035:has been listed at
847:
778:10.21105/joss.01073
2832:
2812:
2751:
2719:
2667:
2626:
2567:
2538:
2474:
2441:
2362:definite integrals
2330:
2310:
2290:
2270:
2241:
2221:
2201:
2172:
2139:
2088:
1936:
1832:
1788:
1675:
1642:
1621:
1597:
1577:
1538:
1492:
1310:
1240:vingt-unième leçon
972:
931:
884:
833:
576:Updated 2007-11-09
462:Mathematics portal
406:content assessment
341:Article milestones
99:dispute resolution
60:
2835:{\displaystyle n}
2742:
2597:
2508:(2003): 155-195.)
2313:{\displaystyle S}
2244:{\displaystyle s}
2231:, the total size
1633:definite integral
869:
822:Formal definition
673:
672:
667:
666:
527:
526:
523:
522:
519:
518:
375:
374:
371:
370:
295:
294:
259:
258:
79:Assume good faith
56:
27:
26:
2959:
2841:
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2821:
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2813:
2808:
2807:
2783:
2782:
2764:
2763:
2750:
2728:
2726:
2725:
2722:{\displaystyle }
2720:
2715:
2714:
2702:
2701:
2676:
2674:
2673:
2668:
2635:
2633:
2632:
2627:
2598:
2596:
2585:
2576:
2574:
2573:
2568:
2547:
2545:
2544:
2539:
2483:
2481:
2480:
2475:
2454:
2449:
2399:Fubini's theorem
2387:
2339:
2337:
2336:
2331:
2319:
2317:
2316:
2311:
2299:
2297:
2296:
2291:
2279:
2277:
2276:
2271:
2250:
2248:
2247:
2242:
2230:
2228:
2227:
2222:
2210:
2208:
2207:
2202:
2181:
2179:
2178:
2173:
2152:
2147:
2097:
2095:
2094:
2089:
2087:
2086:
2082:
2081:
2072:
2050:
2049:
2048:
2047:
1960:
1953:
1947:
1945:
1943:
1942:
1937:
1932:
1931:
1919:
1918:
1897:
1896:
1874:
1863:
1841:
1839:
1838:
1833:
1805:
1799:
1797:
1795:
1794:
1789:
1787:
1774:
1684:
1682:
1681:
1676:
1655:
1650:
1630:
1628:
1627:
1622:
1606:
1604:
1603:
1598:
1586:
1584:
1583:
1578:
1547:
1545:
1544:
1539:
1537:
1501:
1499:
1498:
1493:
1488:
1487:
1475:
1474:
1453:
1452:
1436:
1425:
1395:
1394:
1376:
1375:
1357:
1356:
1319:
1317:
1316:
1311:
1309:
1175:Riemann integral
1154:
1152:
1140:
1134:
1132:
1120:
1057:
1034:
1028:
981:
979:
978:
973:
971:
970:
940:
938:
937:
932:
921:
920:
904:
899:
883:
846:
841:
780:
758:
737:Open Mathematics
720:
714:
708:
702:
601:Article requests
590:
583:
582:
577:
540:
539:
529:
495:
494:
491:
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435:
427:
420:
403:
394:
393:
386:
385:
377:
357:October 23, 2006
338:
304:
297:
287:is not reached,
268:
267:
261:
253:
239:
238:
229:
192:
191:
177:
108:Article policies
29:
16:
2967:
2966:
2962:
2961:
2960:
2958:
2957:
2956:
2897:
2896:
2824:
2823:
2793:
2768:
2755:
2731:
2730:
2706:
2687:
2679:
2678:
2644:
2643:
2589:
2579:
2578:
2550:
2549:
2521:
2520:
2436:
2435:
2403:Stokes' theorem
2322:
2321:
2302:
2301:
2282:
2281:
2253:
2252:
2233:
2232:
2213:
2212:
2184:
2183:
2134:
2133:
2073:
2054:
2039:
2031:
2011:
2010:
1965:
1964:
1963:
1954:
1950:
1923:
1904:
1882:
1844:
1843:
1809:
1808:
1806:
1802:
1778:
1777:
1775:
1771:
1637:
1636:
1613:
1612:
1589:
1588:
1554:
1553:
1504:
1503:
1479:
1460:
1444:
1386:
1367:
1348:
1322:
1321:
1276:
1275:
1215:§ Formalization
1207:Cours d'Analyse
1183:Cours d'Analyse
1157:
1146:
1141:
1137:
1126:
1121:
1117:
1084:Bishop Berkeley
1072:
1051:
1030:
1022:
962:
957:
956:
912:
828:
827:
824:
763:
730:
724:
723:
715:
711:
703:
699:
684:
669:
668:
663:
551:
537:
492:
489:
486:
483:
482:
460:
453:
433:
404:on Knowledge's
401:
391:
289:other solutions
255:
254:
249:
226:
134:
129:
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127:
104:
74:
12:
11:
5:
2965:
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2869:
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2806:
2803:
2800:
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2762:
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2754:
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2741:
2738:
2718:
2713:
2709:
2705:
2700:
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2663:
2660:
2657:
2654:
2651:
2640:
2637:
2625:
2622:
2619:
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2613:
2610:
2607:
2604:
2601:
2595:
2592:
2588:
2566:
2563:
2560:
2557:
2548:is a function
2537:
2534:
2531:
2528:
2513:
2512:
2511:
2510:
2509:
2498:
2491:
2490:
2489:
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2473:
2470:
2467:
2464:
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2458:
2453:
2448:
2444:
2428:
2421:
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2374:
2370:
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2342:
2341:
2329:
2309:
2289:
2269:
2266:
2263:
2260:
2240:
2220:
2200:
2197:
2194:
2191:
2171:
2168:
2165:
2162:
2159:
2156:
2151:
2146:
2142:
2127:
2126:
2125:
2115:
2114:
2113:
2106:
2105:
2104:
2085:
2080:
2076:
2071:
2067:
2064:
2061:
2057:
2053:
2046:
2042:
2038:
2034:
2030:
2027:
2024:
2021:
2018:
2003:
2002:
2001:
1994:
1973:
1962:
1961:
1948:
1935:
1930:
1926:
1922:
1917:
1914:
1911:
1907:
1903:
1900:
1895:
1892:
1889:
1885:
1881:
1878:
1873:
1870:
1867:
1862:
1859:
1856:
1852:
1831:
1828:
1825:
1822:
1819:
1816:
1800:
1786:
1768:
1767:
1763:
1762:
1761:
1760:
1759:
1746:
1745:
1744:
1740:
1734:
1732:
1731:
1728:ruler function
1706:
1705:
1704:
1692:
1691:
1690:
1686:
1674:
1671:
1668:
1665:
1662:
1659:
1654:
1649:
1645:
1620:
1596:
1576:
1573:
1570:
1567:
1564:
1561:
1536:
1532:
1529:
1526:
1523:
1520:
1517:
1514:
1511:
1491:
1486:
1482:
1478:
1473:
1470:
1467:
1463:
1459:
1456:
1451:
1447:
1443:
1440:
1435:
1432:
1429:
1424:
1421:
1418:
1414:
1410:
1407:
1404:
1401:
1398:
1393:
1389:
1385:
1382:
1379:
1374:
1370:
1366:
1363:
1360:
1355:
1351:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1308:
1304:
1301:
1298:
1295:
1292:
1289:
1286:
1283:
1267:
1264:
1263:
1262:
1248:
1211:infinitesimals
1189:
1186:
1156:
1155:
1135:
1114:
1071:
1068:
1021:
1011:
1010:
1009:
995:
969:
965:
930:
927:
924:
919:
915:
911:
908:
903:
898:
895:
892:
888:
882:
879:
876:
872:
868:
865:
862:
859:
856:
853:
850:
845:
840:
836:
823:
820:
819:
818:
782:
781:
760:
759:
743:(1): 983–995,
726:
722:
721:
709:
696:
695:
683:
680:
678:
675:
671:
670:
665:
664:
662:
661:
660:
659:
645:
634:
629:
622:
594:
592:
591:
579:
534:
532:
525:
524:
521:
520:
517:
516:
505:
499:
498:
496:
479:the discussion
466:
465:
449:
437:
436:
428:
416:
415:
409:
387:
373:
372:
369:
368:
365:
358:
354:
353:
350:
347:
343:
342:
334:
333:
305:
293:
292:
269:
257:
256:
247:
245:
244:
241:
240:
194:
193:
131:
130:
126:
125:
120:
115:
106:
105:
103:
102:
95:
90:
81:
75:
73:
72:
61:
52:
51:
48:
47:
41:
25:
24:
19:
13:
10:
9:
6:
4:
3:
2:
2964:
2953:
2950:
2948:
2945:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2915:
2913:
2910:
2908:
2905:
2904:
2902:
2883:
2879:
2875:
2871:
2865:
2864:
2862:
2858:
2855:
2851:
2844:
2843:
2829:
2804:
2801:
2798:
2794:
2787:
2779:
2776:
2773:
2769:
2765:
2760:
2756:
2747:
2743:
2739:
2736:
2711:
2707:
2703:
2698:
2695:
2692:
2688:
2661:
2655:
2652:
2649:
2641:
2638:
2620:
2614:
2611:
2605:
2599:
2593:
2590:
2586:
2561:
2555:
2532:
2526:
2517:
2516:
2514:
2507:
2503:
2502:
2499:
2495:
2494:
2492:
2486:
2471:
2468:
2462:
2456:
2451:
2446:
2442:
2432:
2431:
2429:
2426:
2422:
2419:
2418:
2417:
2413:
2409:
2405:
2404:
2400:
2396:
2392:
2385:
2379:
2375:
2371:
2367:
2363:
2359:
2358:
2357:
2353:
2349:
2344:
2327:
2307:
2287:
2264:
2258:
2238:
2218:
2195:
2189:
2169:
2166:
2160:
2154:
2149:
2144:
2140:
2131:
2130:
2128:
2123:
2119:
2118:
2116:
2110:
2109:
2107:
2102:
2078:
2074:
2069:
2065:
2059:
2055:
2051:
2044:
2040:
2036:
2032:
2028:
2022:
2016:
2007:
2006:
2004:
1998:
1997:
1995:
1992:
1991:
1990:
1986:
1982:
1978:
1974:
1971:
1970:
1969:
1968:
1967:
1966:
1958:
1952:
1949:
1928:
1924:
1920:
1915:
1912:
1909:
1905:
1893:
1890:
1887:
1883:
1876:
1871:
1868:
1865:
1860:
1857:
1854:
1850:
1826:
1823:
1820:
1814:
1804:
1801:
1773:
1770:
1766:
1758:
1754:
1750:
1747:
1741:
1737:
1736:
1735:
1729:
1724:
1720:
1716:
1713:
1710:
1709:
1707:
1702:
1698:
1697:
1696:
1693:
1687:
1672:
1669:
1663:
1657:
1652:
1647:
1643:
1634:
1618:
1610:
1594:
1571:
1568:
1565:
1559:
1552:the quantity
1551:
1524:
1521:
1518:
1512:
1509:
1484:
1480:
1476:
1471:
1468:
1465:
1461:
1449:
1445:
1438:
1433:
1430:
1427:
1422:
1419:
1416:
1412:
1408:
1399:
1396:
1391:
1387:
1383:
1380:
1377:
1372:
1368:
1364:
1361:
1358:
1353:
1349:
1342:
1339:
1336:
1333:
1327:
1296:
1293:
1290:
1284:
1281:
1273:
1272:
1271:
1268:
1265:
1261:
1257:
1253:
1249:
1246:
1242:
1241:
1236:
1235:
1230:
1229:
1228:
1224:
1220:
1216:
1212:
1208:
1204:
1203:
1202:
1198:
1194:
1190:
1187:
1184:
1180:
1176:
1172:
1171:
1170:
1169:
1165:
1161:
1150:
1144:
1139:
1136:
1130:
1125:, pp. 628–629
1124:
1119:
1116:
1113:
1109:
1104:
1099:
1097:
1093:
1089:
1085:
1081:
1075:
1069:
1067:
1066:
1062:
1058:
1056:
1055:
1048:
1047:
1042:
1038:
1033:
1029:The redirect
1027:
1020:
1016:
1012:
1008:
1004:
1000:
996:
993:
989:
985:
967:
963:
954:
953:
952:
951:
947:
943:
942:211.30.47.108
928:
917:
913:
906:
901:
896:
893:
890:
886:
874:
866:
863:
860:
854:
848:
843:
838:
834:
821:
817:
813:
809:
805:
804:
803:
802:
798:
794:
790:
786:
779:
775:
771:
767:
762:
761:
757:
754:
750:
746:
742:
738:
734:
729:
728:
727:
718:
713:
710:
706:
701:
698:
694:
692:
687:
681:
679:
676:
658:
654:
652:
651:
646:
643:
641:
640:
635:
633:
630:
627:
623:
621:
618:
617:
616:
612:
609:
605:
603:
602:
597:
596:
593:
589:
585:
584:
581:
578:
575:
572:
569:
566:
563:
560:
557:
554:
550:
548:
544:
533:
531:
530:
514:
510:
504:
501:
500:
497:
480:
476:
472:
471:
463:
457:
452:
450:
447:
443:
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161:free images
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2901:Categories
2364:, and the
1776:I'm using
1765:References
543:To-do list
367:Not listed
2488:function.
1739:details).
1143:Katz 2009
1123:Katz 2009
1054:Steel1943
756:2391-5455
400:is rated
285:consensus
101:if needed
84:Be polite
34:talk page
2408:D.Lazard
1699:I found
1219:D.Lazard
1145:, p. 785
639:Copyedit
626:Cronholm
547:Integral
308:Integral
199:Archives
69:get help
42:This is
40:article.
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2425:defined
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2384:Riemann
2373:Cayley.
2348:LambdaP
2117:Later:
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559:history
511:on the
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1193:Mgnbar
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650:Expand
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