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History of calculus
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====Middle East==== {{See also|Mathematics in the medieval Islamic world}} [[file:Ibn al-Haytham crop.jpg|thumb|upright=.7|Ibn al-Haytham, 11th-century Arab mathematician and physicist]] In the Middle East, [[Ibn al-Haytham|Hasan Ibn al-Haytham]], Latinized as Alhazen ({{c.|lk=no|965|1040}} AD) derived a formula for the sum of [[fourth power]]s. He 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 [[Integral|integration]], where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a [[paraboloid]].<ref name=katz>{{Cite journal|last=Katz |first=Victor J. |author-link=Victor J. Katz |date=June 1995 |title=Ideas of Calculus in Islam and India |url=https://www.tandfonline.com/doi/full/10.1080/0025570X.1995.11996307 |journal=[[Mathematics Magazine]] |language=en |volume=68 |issue=3 |pages=163–174 |doi=10.1080/0025570X.1995.11996307 |issn=0025-570X |jstor=2691411}}</ref> [[Roshdi Rashed]] has argued that the 12th century mathematician [[Sharaf al-Dīn al-Tūsī]] must have used the derivative of cubic polynomials in his ''Treatise on Equations''. Rashed's conclusion has been contested by other scholars, who argue that he could have obtained his results by other methods which do not require the derivative of the function to be known.<ref>{{cite journal |last1=Berggren |first1=J. L. |last2=Al-Tūsī |first2=Sharaf Al-Dīn |last3=Rashed |first3=Roshdi |last4=Al-Tusi |first4=Sharaf Al-Din |title=Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's Muʿādalāt |journal=[[Journal of the American Oriental Society]] |date=April 1990 |volume=110 |issue=2 |pages=304–309 |doi=10.2307/604533|jstor=604533 }}</ref>
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