Editing Integral (section)

== History ==
{{See also|History of calculus}}

=== Pre-calculus integration ===
The first documented systematic technique capable of determining integrals is the [[method of exhaustion]] of the [[Ancient Greece|ancient Greek]] astronomer [[Eudoxus of Cnidus|Eudoxus]] and philosopher [[Democritus]] (''ca.'' 370 BC), which sought to find areas and volumes by breaking them up into an infinite number of divisions for which the area or volume was known.<ref>{{Harvnb|Burton|2011|p=117}}.</ref> This method was further developed and employed by [[Archimedes]] in the 3rd century BC and used to calculate the [[area of a circle]], the [[surface area]] and [[volume]] of a [[sphere]], area of an [[ellipse]], the area under a [[parabola]], the volume of a segment of a [[paraboloid]] of revolution, the volume of a segment of a [[hyperboloid]] of revolution, and the area of a [[spiral]].<ref>{{Harvnb|Heath|2002}}.</ref>

A similar method was independently developed in [[China]] around the 3rd century AD by [[Liu Hui]], who used it to find the area of the circle. This method was later used in the 5th century by Chinese father-and-son mathematicians [[Zu Chongzhi]] and [[Zu Geng (mathematician)|Zu Geng]] to find the volume of a sphere.<ref>{{harvnb|Katz|2009|pp=201–204}}.</ref>

In the Middle East, Hasan Ibn al-Haytham, Latinized as [[Alhazen]] ({{c.|965|lk=no|1040}}&nbsp;AD) derived a formula for the sum of [[fourth power]]s.<ref>{{harvnb|Katz|2009|pp=284–285}}.</ref> Alhazen determined the equations to calculate the area enclosed by the curve represented by <math>y=x^k</math> (which translates to the integral <math>\int x^k \, dx</math> in contemporary notation), for any given non-negative integer value of <math>k</math>.<ref>{{Cite journal |last=Dennis |first=David |last2=Kreinovich |first2=Vladik |last3=Rump |first3=Siegfried M. |date=1998-05-01 |title=Intervals and the Origins of Calculus |url=https://doi.org/10.1023/A:1009989211143 |journal=Reliable Computing |language=en |volume=4 |issue=2 |pages=191–197 |doi=10.1023/A:1009989211143 |issn=1573-1340}}</ref> He used the results to carry out what would now be called an integration of this function, where the formulae for the sums of integral squares and fourth powers allowed him to calculate the volume of a [[paraboloid]].<ref>{{harvnb|Katz|2009|pp=305–306}}.</ref>

The next significant advances in integral calculus did not begin to appear until the 17th century. At this time, the work of [[Bonaventura Cavalieri|Cavalieri]] with his [[method of indivisibles]], and work by [[Pierre de Fermat|Fermat]], began to lay the foundations of modern calculus,<ref>{{harvnb|Katz|2009|pp=516–517}}.</ref> with Cavalieri computing the integrals of {{math|''x''<sup>''n''</sup>}} up to degree {{math|''n'' {{=}} 9}} in [[Cavalieri's quadrature formula]].<ref>{{Harvnb|Struik|1986|pp=215–216}}.</ref> The case ''n'' = &minus;1 required the invention of a [[function (mathematics)|function]], the [[hyperbolic logarithm]], achieved by [[quadrature (mathematics)|quadrature]] of the [[hyperbola]] in 1647.

Further steps were made in the early 17th century by [[Isaac Barrow|Barrow]] and [[Evangelista Torricelli|Torricelli]], who provided the first hints of a connection between integration and [[Differential calculus|differentiation]]. Barrow provided the first proof of the [[fundamental theorem of calculus]].<ref>{{harvnb|Katz|2009|pp=536–537}}.</ref> [[John Wallis|Wallis]] generalized Cavalieri's method, computing integrals of {{mvar|x}} to a general power, including negative powers and fractional powers.<ref>{{Harvnb|Burton|2011|pp=385–386}}.</ref>

=== Leibniz and Newton ===
The major advance in integration came in the 17th century with the independent discovery of the [[fundamental theorem of calculus]] by [[Gottfried Wilhelm Leibniz|Leibniz]] and [[Isaac Newton|Newton]].<ref>{{Harvnb|Stillwell|1989|p=131}}.</ref> The theorem demonstrates a connection between integration and differentiation. This connection, combined with the comparative ease of differentiation, can be exploited to calculate integrals. In particular, the fundamental theorem of calculus allows one to solve a much broader class of problems. Equal in importance is the comprehensive mathematical framework that both Leibniz and Newton developed. Given the name infinitesimal calculus, it allowed for precise analysis of functions with continuous domains. This framework eventually became modern [[calculus]], whose notation for integrals is drawn directly from the work of Leibniz.

=== Formalization ===
While Newton and Leibniz provided a systematic approach to integration, their work lacked a degree of [[Rigor#Mathematical rigour|rigour]]. [[George Berkeley|Bishop Berkeley]] memorably attacked the vanishing increments used by Newton, calling them "[[The Analyst#Content|ghosts of departed quantities]]".<ref>{{harvnb|Katz|2009|pp=628–629}}.</ref> Calculus acquired a firmer footing with the development of [[Limit (mathematics)|limits]]. Integration was first rigorously formalized, using limits, by [[Bernhard Riemann|Riemann]].<ref>{{harvnb|Katz|2009|p=785}}.</ref> Although all bounded [[piecewise]] continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of [[Fourier analysis]]—to which Riemann's definition does not apply, and [[Henri Lebesgue|Lebesgue]] formulated a [[#Lebesgue integral|different definition of integral]], founded in [[Measure (mathematics)|measure theory]] (a subfield of [[real analysis]]). Other definitions of integral, extending Riemann's and Lebesgue's approaches, were proposed. These approaches based on the real number system are the ones most common today, but alternative approaches exist, such as a definition of integral as the [[standard part]] of an infinite Riemann sum, based on the [[hyperreal number]] system.

=== Historical notation ===
The notation for the indefinite integral was introduced by [[Gottfried Wilhelm Leibniz]] in 1675.<ref>{{Harvnb|Burton|2011|loc=p.&nbsp;414}}; {{Harvnb|Leibniz|1899|loc=p.&nbsp;154}}.</ref> He adapted the [[integral symbol]], '''∫''', from the letter ''ſ'' ([[long s]]), standing for ''summa'' (written as ''ſumma''; Latin for "sum" or "total"). The modern notation for the definite integral, with limits above and below the integral sign, was first used by [[Joseph Fourier]] in ''Mémoires'' of the French Academy around 1819–1820, reprinted in his book of 1822.<ref>{{Harvnb|Cajori|1929|loc=pp.&nbsp;249–250}}; {{Harvnb|Fourier|1822|loc=§231}}.</ref>

[[Isaac Newton]] used a small vertical bar above a variable to indicate integration, or placed the variable inside a box. The vertical bar was easily confused with {{math|{{overset|'''.'''|''x''}}}} or {{math|''x''′}}, which are used to indicate differentiation, and the box notation was difficult for printers to reproduce, so these notations were not widely adopted.<ref>{{Harvnb|Cajori|1929|p=246}}.</ref>

=== First use of the term ===
The term was first printed in Latin by [[Jacob Bernoulli]] in 1690: "Ergo et horum Integralia aequantur".<ref>{{Harvnb|Cajori|1929|p=182}}.</ref>