Talk:Integral

The redirect Integration with other techniques has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2024 January 31 § Integration with other techniques until a consensus is reached. Steel1943 (talk) 21:04, 31 January 2024 (UTC)Reply

In the History section, the subsection Formalization begins with:

While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of rigour. Bishop Berkeley memorably attacked the vanishing increments used by Newton, calling them "ghosts of departed quantities".^[1] Calculus acquired a firmer footing with the development of limits. Integration was first rigorously formalized, using limits, by Riemann.^[2]

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:

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 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.

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?

LambdaP (talk) 14:39, 3 May 2024 (UTC)Reply

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 Riemann integral (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 Augustin-Louis Cauchy or Cours d'Analyse.

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.

Once we understand that, if everything holds up, then multiple Wikipedia articles will need to be changed. Mgnbar (talk) 16:09, 3 May 2024 (UTC)Reply

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 Cours d'Analyse, he used the informal (at that time) concept of infinitesimals. 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 § Formalization does not require to be changed. D.Lazard (talk) 16:42, 3 May 2024 (UTC)Reply

Right. I went and read Cauchy’s Résumé des leçons données à l'École royale polytechnique sur le calcul infinitésimal, which was published in 1823. I think the relevant part is in the vingt-unième leçon, starting on p. 81. On p. 83, the same leçon includes an explicit discussion that the way of cutting intervals does not change the limit value of the integral.

With respect to the infinitesimals, it's less clear, but the word doesn't seem to appear in the proof. LambdaP (talk) 20:54, 3 May 2024 (UTC)Reply

To specifically answer some of your points:

Definition of the integral

Cauchy seems to have been the first to define the integral of a function ${\displaystyle f:[a,b]\to \mathbb {R} }$^[1] using the quantity ${\displaystyle S(f,\sigma =(x_{0}=a,x_{1},\ldots ,x_{n}=b))=\sum _{i=0}^{n-1}f(x_{i})(x_{i+1}-x_{i})}$, 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 ${\displaystyle f:[a,b]\to \mathbb {R} }$ is uniformly continuous), that a) the quantity ${\displaystyle S(f,\sigma )}$ converges to some limit value as the mesh of ${\displaystyle \sigma }$ tends to zero, and b) this limit value does not depend on the choice of partitions ${\displaystyle \sigma }$. He calls this limit value a definite integral, which he suggests we write ${\displaystyle \int _{a}^{b}f(x)dx}$, in passing saying this notation was “imagined by Mr. Fourier”^[2].

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^[3] On the other hand, Cauchy seemed mainly interested in actually integrating functions, rather than studying 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.

Riemann's contribution

I found a good StackExchange answer 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:

Riemann's nontrivial contributions to this topic were: (A) giving a necessary and sufficient condition for integrability based on the behavior of a function; (B) using this condition to prove the integrability of a certain function having a dense set of discontinuities; (C) 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 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 ruler function, which is also called the Thomae function because it first appeared in an 1875 booklet by Thomae.

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 details).

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 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.

LambdaP (talk) 15:43, 4 May 2024 (UTC)Reply

This is all great to read. Thanks for putting it together.

But I worry that we're straying into Wikipedia:No original research, 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. Mgnbar (talk) 15:53, 4 May 2024 (UTC)Reply

I agree. I personally think we have everything we need in (Katz 2009).

Chapter 22 is titled “Analysis in the Nineteenth Century”. In the chapter introduction (p. 765), we read:

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. Riemann further developed and extended these ideas in the middle of the century.

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:

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 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 ${\displaystyle f(x)=e^{-x^{2}}+e^{-(1/x^{2})}}$ 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 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, Résumé des leçons données à l’École Royale Polytechnique sur le calcul infinitesimal, for it is these texts, used in Paris, that provided the model for calculus texts for the remainder of the century.

Section 22.1.5 opens with:

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 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.

Later:

In the second part of his Résumé, 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.

Section 22.1.6 discusses Fourier's work, then in Section 22.1.7 (“The Riemann Integral”):

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 ${\displaystyle \int _{a}^{b}f(x)dx}$.… 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 ${\displaystyle f(x)}$ to be integrable: “If, with the infinite decrease of all the quantities ${\displaystyle \delta }$, the total size ${\displaystyle s}$ of the intervals in which the variations of the function ${\displaystyle f(x)}$ are greater than a given quantity ${\displaystyle \sigma }$ always becomes infinitely small in the end, then the sum ${\displaystyle S}$ converges when all the ${\displaystyle \delta }$ become infinitely small” and conversely.

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. LambdaP (talk) 17:42, 4 May 2024 (UTC)Reply

It seems that Katz did a confusion between "definition" and "computation": If the above quotation would be taken literaly, this would mean that definite integrals, and the fundamental theorem of calculus 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 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.

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 Cayley.

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 century, and thus that Cayley did not have a formal definition of the real numbers used in its "formal" definition of integrals).

For these reasons, I suggest to theplace the sentence "Integration was first rigorously formalized, using limits, by Riemann" with Riemann was the first to provide a coherent theory of integration that includes as corollaries the fundamental theorem of calculus as well as other fundamental theorems such as Fubini's theorem and Stokes' theorem. D.Lazard (talk) 13:16, 5 May 2024 (UTC)Reply

I presume you mean Cauchy rather than Cayley? :-)

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 defined the integral as the limit of a sum (see above quotes). Here's a few other secondary sources saying the same thing.

Here's Lützen (p. 170):

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 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 ${\displaystyle \int _{a}^{b}f(x)dx}$ (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).

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 function.

further down (pp. 171-172):

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 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… [a]nd most importantly, Cauchy changed the technique from being a numerical approximation procedure to being a definition....

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 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).

(Lützen, Jesper. "The foundation of analysis in the 19th century." A history of analysis (2003): 155-195.)

Here's Gray (p. 55):

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 ${\displaystyle f(x)}$ is a function ${\displaystyle F(x)}$ with the property that ${\displaystyle {\frac {d}{dx}}F(x)=f(x)}$. As such, it is determined up to an additive constant.

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.

The Cauchy integral of a function of a real variable ${\displaystyle y=f(x)}$ that is continuous in a given interval was defined as follows. He divided the interval into n equal subintervals ${\displaystyle [x_{i-1},x_{i}]}$, and considered the sum ${\displaystyle S=\sum _{i}(x_{i}-x_{i-1})f(x_{i-1})}$. The integral will be the limit of this sum as ${\displaystyle n}$ tends to infinity.

(Gray, Jeremy. The real and the complex: a history of analysis in the 19th century. Cham: Springer, 2015.)

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 Résumé (see e.g., Katz p. 778 or Lützen p. 171). Not sure about Fubini or Stokes' theorems, it's an interesting question for sure.

As an aside, shortly after he published his Résumé, Cauchy was scolded by Polytechnique for focusing too much on rigor and not on practical, engineering-oriented stuff. Quoting Katz again (p. 779):

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 point, but there is evidence that Cauchy was reproached by the directors of the school. He was told that, because the École Polytechnique [sic] 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.

I doubt practical matters of engineering was a strong driver of how he shaped his course prior to that point. LambdaP (talk) 21:25, 5 May 2024 (UTC)Reply

Really, blue and yellow for the integral drawing? being a mathematician doesn't mean that you have to do horrible stylistic choices. Was it so difficult to stick to conventions like green and red for positive and negative? Whateverowo (talk) 18:14, 5 December 2024 (UTC)Reply

I have never heard of the red-green convention that you mention. Meanwhile, there are issues around color blindness to consider. So far, I am not convinced. Cheers, Mgnbar (talk) 18:21, 5 December 2024 (UTC)Reply

Yeah, and there are people with tritanopia that wouldn't differentiate blue/yellow colours either, there's also no mention of choosing colours to help colour blind people in the description and furthermore, they would be useless because of the plus and minus symbols. Again,these are just poor colour choices, and colours here are there for aesthetic reasons. Whateverowo (talk) 20:12, 5 December 2024 (UTC)Reply

Trianopia* Google keyboard things Whateverowo (talk) 20:24, 5 December 2024 (UTC)Reply

And if colour blindness is the issue, why not use red-blue pair that's way nicer and still colour blind friendly. Whateverowo (talk) 20:40, 5 December 2024 (UTC)Reply

And really, you have never seen graphs of the stock market, a spreadsheet? Have you heard also of green and red flags? There's this whole notion in culture where the green-red pair is associated with good and bad things, positive and negative things. Whateverowo (talk) 20:55, 5 December 2024 (UTC)Reply

There's studies about students getting more anxiety with grades with red ink rather than with green ink, fast foods chains use red to give people a sense of urgency and in Europe that generally that's not as liked, they charge red with green. It really runs that deep culturally Whateverowo (talk) 21:01, 5 December 2024 (UTC)Reply

It should be blue and red Hu741f4 (talk) 21:41, 5 December 2024 (UTC)Reply

Yes, I have seen a blue and red color convention. I would support changing to that. Cheers, Mgnbar (talk) 22:48, 5 December 2024 (UTC)Reply

Glad this came to a productive resolution :) Whateverowo (talk) 22:56, 5 December 2024 (UTC)Reply

Overall this is one of the best illustrated math articles. Whateverowo: can you please try to be significantly more respectful of image authors? I find your tone here quite insulting.

More substantively: Unless very carefully done, red and green is generally a miserable combination for this kind of thing, for legibility reasons. The specific hues chosen don't matter too much, but there needs to be significant lightness contrast between the two colors. The current blue and yellow picture is well done and very legible, and I support keeping it. There are some color changes I would recommend though: the colors chosen for File:Riemann Integration and Darboux Upper Sums.gif (etc.), File:Lebesgueintegralsimplefunctions finer-dotted.svg, and File:Surface_integral_illustration.svg are all somewhat distractingly intense. Moderately reducing the colorfulness of these would be an improvement. –jacobolus (t) 14:55, 6 December 2024 (UTC)Reply

I never was talking about legibility, I was talking about aesthetics, yes the images do illustrate the concepts very well as they are now, and also I did change my mind from green and red to blue and red, to be more aesthetically pleasing and still readable for colourblind people.

I don't think however that the images you have put are distracting at all, they do their job just fine :) Whateverowo (talk) 14:32, 8 December 2024 (UTC)Reply

If it came as insulting, I'm sorry, that wasn't never my objective. Whateverowo (talk) 14:33, 8 December 2024 (UTC)Reply

Came of* Whateverowo (talk) 14:33, 8 December 2024 (UTC)Reply

The current image is aesthetically very well put together, with good use of space, thick lines, clear labels, and a very legible color scheme (the colors currently in use are close to ideally separated under any kind of color vision deficiency, with significant lightness contrast and colors which are substantially distinct to all of the cone cells in the eye). I would not recommend replacing it, and would be outright opposed to any replacement which was not made extremely carefully. –jacobolus (t) 16:57, 8 December 2024 (UTC)Reply

I'm talking about a hue change and I've spent enough time looking at diagrams for colour blindness, changing the yellow to #F44444 would only change the hue, not saturation or value (: Whateverowo (talk) 22:51, 9 December 2024 (UTC)Reply

I would recommend against even making hue changes here; the current colors were well selected and seem entirely fine to me. The complaint about cultural associations of red with negative seems exaggerated, quite culturally specific (e.g. Chinese readers have completely opposite associations with red), and generally misguided to me. spent enough time looking at diagrams for colour blindness – then you should probably realize that the blue and yellow colors currently used in this diagram are very clearly distinguishable for folks with Protanopia, Deuteranopia, Tritanopia, or even Achromatopsia. –jacobolus (t) 23:28, 9 December 2024 (UTC)Reply

What does that mean? ~2026-37657-79 (talk) 21:30, 30 June 2026 (UTC)Reply

An integral is analogous to a sum. But whereas a sum is a discrete set of numbers added together, an integral is (loosely speaking) a continuous set of numbers added together. Does that help? How much should the article clarify on this point? Mgnbar (talk) 21:48, 30 June 2026 (UTC)Reply

In Integral § Pre-calculus integration, do we actually need the parenthetical saying (which translates to the integral ${\displaystyle \int x^{k}\,dx}$ in contemporary notation)? I don't think so: to me, it just looks like throwing in anachronistic notation that the article hasn't introduced yet, which doesn't really help anybody. My removal was reverted with the rationale "the source mentions this", which seems insufficient to me. We don't include things just because a source mentions them. And it is only a mention, a parenthetical in the source that is nearly all about someone else (Wallis, not Alhazen). As I see it, this fails WP:DUE. Stepwise Continuous Dysfunction (talk) 06:29, 29 July 2026 (UTC)Reply

That doesn't mean we should omit some parts from the source and present only half of the information. This is WP:CHERRYPICKING and WP:SYNTH. The cited source explicitly mentions this notation in relation to Alhazen. The content says, "(which translates to the integral ${\displaystyle \int x^{k},dx}$ in contemporary notation)." You wrote that this is "anachronistic" and "doesn't really help anybody." It is the exact opposite. Translating historical geometric achievements into contemporary calculus notation is exactly how academic historians make these concepts clear to modern readers. The peer-reviewed source, Intervals and the Origins of Calculus by David Dennis et al., uses this exact framing, stating that Alhazen found the formulas for the area bounded by the curve ${\displaystyle y=x^{k}}$, and specifically adds "(in modern terms, ${\displaystyle \int x^{k}dx}$)". Retaining the parenthetical simply reflects the source's own explanatory method. The source is about the development of calculus. Your justification for WP:DUE is inadequate because other sources, like Victor J. Katz (Katz, Victor J. (2009), A History of Mathematics: An Introduction, Addison-Wesley), support the claim. WP:DUE applies to minority views that are rejected by other sources. The fact that Dennis used C. H. Edwards as a source for this claim means he isn't representing a minority view. Alhazen's calculation of the area bounded by a curve represented by ${\displaystyle y=x^{k}}$ is supported by other reliable sources. The authors explicitly identify his integration work as "one of the first breakthroughs" in calculus. Alhazen's explicit polynomial formulas provided the direct foundation that John Wallis later used to construct his interpolation tables. Retaining the modern notation accurately reflects the source material and helps readers.

Edit: Historian of Mathematics Roshdi Rashed mentions this notation too but with limits:

Moreover, Ibn al-Haytham then uses the inequalities ${\displaystyle I_{n}<v<C_{n}}$ and shows that, for all ${\displaystyle \varepsilon >0}$, there exists an ${\displaystyle N}$ such that, for ${\displaystyle n\geq N}$, we have ${\displaystyle v-I_{n}<\varepsilon ,C_{n}-v<\varepsilon ,}$ which proves that ${\displaystyle I_{n}}$ tends to ${\displaystyle v}$ and similarly for ${\displaystyle C_{n}}$. Thus we certainly have ${\displaystyle v=\int _{0}^{R}f(x)dx.}$ In other words, Ibn al-Haytham's calculation is equivalent to a simple Cauchy-Riemann integral.^[1]

Hu741f4 (talk) 13:32, 29 July 2026 (UTC)Reply

The author says his ideas of integration is one of the major breakthrough of calculus is completely wrong.The idea of integration goes way back during the time of archimedes.Archimedes method of exhaustion is considered as integration although he didn't used modern integration method or symbol like newton or leibniz so did ibn al hayatam,the usage of integral as a symbol of usage during modern day method came during leibniz or Newton works (leibniz symbol is used in modern calculus) but associating ibn al hayatam method to modern day integration symbol is wrong it's like association archimedes method of exhaustion techniqu6to find area of shape to modern integral notation Myuoh kaka roi (talk) 14:23, 29 July 2026 (UTC)Reply

"Goes back" and "breakthrough" are two different things. I am not making things up. This is what sources say. Where are your sources to counter this??? Your characterization of what these sources say regarding Alhazen, as an idea which is "completely wrong", is nothing but your original research WP:OR or personal opinion. Here in Wikipedia, we do not omit contents from sources based on our personal opinion or original research. The cited source says:

Greek mathematics left many problems unsolved, including the problem of finding the areas of even such simple geometric figures as a domain bounded by a hyperbola ${\displaystyle y=1/x(i.e.,y=x^{-1})}$. One of the first breakthroughs came around the year 1000, when the Arabic mathematician Abu Ali al-Hassan bel al-Hassan ben Haitam, or shortly al-Hassan (965–1038), known in the West as Alhazen, found the formulas for the area of a domain bounded by the curve ${\displaystyle y=x^{k}}$ (in modern terms, ${\displaystyle \int x^{k}dx}$) for an arbitrary nonnegative integer k [4], [5]. This discovery enabled Alhazen to compute the areas and volumes of the curves and surfaces bounded by polynomial equations ${\displaystyle y=P(x)}$. ^[2]

Historian of Mathematics Roshdi Rashed mentions this integral too:

Moreover, Ibn al-Haytham then uses the inequalities ${\displaystyle I_{n}<v<C_{n}}$ and shows that, for all ${\displaystyle \varepsilon >0}$, there exists an ${\displaystyle N}$ such that, for ${\displaystyle n\geq N}$, we have ${\displaystyle v-I_{n}<\varepsilon ,C_{n}-v<\varepsilon ,}$ which proves that ${\displaystyle I_{n}}$ tends to ${\displaystyle v}$ and similarly for ${\displaystyle C_{n}}$. Thus we certainly have ${\displaystyle v=\int _{0}^{R}f(x)dx.}$ In other words, Ibn al-Haytham's calculation is equivalent to a simple Cauchy-Riemann integral.^[3]

Hu741f4 (talk) 16:57, 29 July 2026 (UTC)Reply

That doesn't mean that he invented integral.Archimedes method of exhaustion is similar to modern day integration both gives the same results in some shapes like circle and parabola but it doesn't equate to other stuff beyond the boundary and his method of exhaustion is more lengthy process not equate to modern day integration nor integral just like that ibn al hayatam find a formula for a area dominated by the curve but he didn't used modern day integration method but used a lengthy method to solve it similar to method of exhaustion as ibn al hayatam wasn't aware of modern day integration nor integral which was developed by Newton or leibniz and secondly using modern equation or symbol for earlier pre calculus integration is odd and require more reliable source to back.Roshif Rashed source is mentioned but no other sources apart from it equate it to modern day integral in terms. Myuoh kaka roi (talk) 17:10, 29 July 2026 (UTC)Reply

ok so where in the article it is mentioned that "Alhazen invented integral"? Or used the integral notation? That is a way how historians of mathematics explain the mathematical work of pre-modern mathematicians to modern readers. Both Dennis and Rashed are simply explaining the work of Alhazen in modern terms. The content perfectly reflects the source by making it clear "(which translates to the integral ${\displaystyle \int x^{k},dx}$ in contemporary notation)." You aren't citing any source to counter this claim or citing a Wikipedia policy that this content violate. What you are presenting is your personal opinion mixed with your original research Hu741f4 (talk) 17:23, 29 July 2026 (UTC)Reply

The issue is representing it in modern terms which is insufficient of what stepwise actually represent.I need to see whether other editors agree with its representation.@Stepwise Continuous Dysfunction@Jacobolus to pin for this matter Myuoh kaka roi (talk) 17:55, 29 July 2026 (UTC)Reply

you are engaging in WP:CANVASSING. You are tagging specific editors and influencing the outcome by making edits on their talk pages in a way that doesn't present the concern in neutral way. This isn't the first time you are doing this. I have enough evidence. Infact the talk page of @Jacobolus is filled with many such edits by you and hence you tagged them again. These users whom you frequently tag and whose talk pages you frequently edit will probably not present a neutral view. Hu741f4 (talk) 20:15, 29 July 2026 (UTC)Reply

Can you link to a direct translation of Ibn al-Haytham's work? He didn't literally say "for all ⁠${\displaystyle \varepsilon >0}$⁠ there exists ...". It's helpful to actually quote and explain the original text rather than only a modernized paraphrase; the latter will be substantially misleading to any reader who interprets it as a quotation. A scholar's book aimed at well-prepared experts can safely assume that they will understand the context, but a Wikipedia article intended for a general audience should be a bit more careful. –jacobolus (t) 21:25, 29 July 2026 (UTC)Reply

Also we should find a better source than one that just mentions this as an aside in passing. Dennis, Kreinovich, & Rump cite Edwards (1979) The Historical Development of the Calculus as their source. We could either go directly to Edwards's book or look for alternate sources discussing this topic in greater detail. –jacobolus (t) 21:33, 29 July 2026 (UTC)Reply

Update: Here's a translation https://www2.kenyon.edu/Depts/Math/Aydin/Teach/Sp24/128/IbnHaythamVolume.pdf –jacobolus (t) 22:23, 29 July 2026 (UTC)Reply

I think it's at least somewhat misleading to imply that this was conceived as the integral of an arbitrary function, per se. It seems clearer to say that Al-Haytham used the method of exhaustion in adding the areas of cylindrical shellsslices, along with some formulas he worked out for the sums of cubes and 4th powers of the first several natural numbers, to calculate the volumes of certain solids of revolution. The method doesn't seem fundamentally different from that of Archimedes (or Eudoxus). –jacobolus (t) 23:30, 29 July 2026 (UTC)Reply

Thanks for finding that translation. It's interesting that at one point, Jan P. Hogendijk comments, "the text is unclear, probably the manuscripts are corrupted, and the editor Rashed does not understand the mathematics".

Edwards writes, "Al-Haitham (ca. 965-1039), known in the West as Alhazen, wrote an influential treatise on geometrical optics and extended some of Archimedes' volume results. For example, he showed that, if a segment of a parabola is revolved about its base (rather than about its axis, as in Archimedes' On Conoids), then the volume of the solid obtained is 8/15 that of the circumscribed cylinder. This computation required formulas for the sums of the first n cubes and fourth powers whereas Archimedes had used only the formulas for the sums of the first n integers and of their squares." I am not sure how this became the claim that is in our article now. Implying that Alhazen was trying to compute what we'd call the integral of an arbitrary function, or an arbitrary integer power of x, seems like Whig history with maybe a a little telephone game going on. Stepwise Continuous Dysfunction (talk) 00:52, 30 July 2026 (UTC)Reply

Judging from another paper by Hogendijk, Hogendijk and Rashed generally have/had a bit of a tiff. Hogendijk's dissertation was a critical edition/translation of a work of Al-Haytham (a reconstruction of the lost last book of Apollonius' Conics), which Rashed later made a different edition/translation of, based on the same manuscript. Hogendijk was unhappy because, among other reasons, he felt Rashed copied many of his non-obvious editorial decisions without appropriate attribution. (In the manuscript, many diagrams and proofs are mathematically incoherent or corrupted because the copyist didn't understand the material and had made mistakes. Someone making a modern edition must try to figure out what these mistakes might have been and correct them, which involves quite a bit of difficult puzzling.) –jacobolus (t) 05:05, 30 July 2026 (UTC)Reply

Maybe we should replace the whole Alhazen bit with a line saying something like, "Alhazen extended Archimedes' results to other solids of revolution". Stepwise Continuous Dysfunction (talk) 18:29, 30 July 2026 (UTC)Reply

Or we could elaborate slightly: we could mention his recursive method for finding the sum of kth powers (which he only applied for k = 2, 4 needed for the problem of finding the volume of a parabola revolved around a line perpendicular to its axis, but using a method which easily generalizes).

In any case, Al-Haytham's work is a nice application of the method of exhaustion using clever geometry and algebra, but it's not the same as a general theory of integrals of polynomials. Katz's History of Mathematics also does a decent job explicitly describing this work and putting it in context. Or for a more specific source, Katz (1995) "Ideas of Calculus in Islam and India"doi:10.1080/0025570X.1995.11996307. –jacobolus (t) 19:20, 30 July 2026 (UTC)Reply

How about this, cited to Katz (1995): "Al-Haytham extended Archimedes' method of exhaustion, finding the volume of a paraboloid formed by rotating a parabola around a line perpendicular to its axis. To achieve this, Al-Haytham invented a way of computing the sums of kth powers, which he applied for the sums of squares and the sums of fourth powers." Stepwise Continuous Dysfunction (talk) 18:06, 31 July 2026 (UTC)Reply

That's not what cherry-picking or synthesis mean. The article should (and does) reflect the facts stated in the sources. It does not have to follow the exact presentation of those facts in regards to mathematical notation. Remember, a Wikipedia article is a summary of the information in the sources, not a sequence of excerpts from them. We can make the decision of whether to include the modern notation based on whether it helps clarify things for the reader, or interrupts the flow of the article. Elestrophe (talk) 19:57, 29 July 2026 (UTC)Reply

Exactly. And using a notation that the article hasn't yet explained does interrupt the flow. Wikipedia has a reputation for being incomprehensible about math. Sometimes that's warranted, and sometimes one could argue that we're doing the best we can with material that's just intrinsically difficult. When we throw in jargon and notation that we don't actually need, we make the material more difficult than necessary. Stepwise Continuous Dysfunction (talk) 21:50, 29 July 2026 (UTC)Reply