Editing Integral (section)

=== 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>