Editing Calculus (section)

=== Fundamental theorem ===
{{Main|Fundamental theorem of calculus}}
The [[fundamental theorem of calculus]] states that differentiation and integration are inverse operations.<ref name=":2" />{{Rp|page=290}} More precisely, it relates the values of antiderivatives to definite integrals. Because it is usually easier to compute an antiderivative than to apply the definition of a definite integral, the fundamental theorem of calculus provides a practical way of computing definite integrals. It can also be interpreted as a precise statement of the fact that differentiation is the inverse of integration.

The fundamental theorem of calculus states: If a function {{math|''f''}} is [[continuous function|continuous]] on the interval {{math|[''a'', ''b'']}} and if {{math|''F''}} is a function whose derivative is {{math|''f''}} on the interval {{math|(''a'', ''b'')}}, then

:<math>\int_{a}^{b} f(x)\,dx = F(b) - F(a).</math>

Furthermore, for every {{math|''x''}} in the interval {{math|(''a'', ''b'')}},

:<math>\frac{d}{dx}\int_a^x f(t)\, dt = f(x).</math>

This realization, made by both [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]], was key to the proliferation of analytic results after their work became known. (The extent to which Newton and Leibniz were influenced by immediate predecessors, and particularly what Leibniz may have learned from the work of [[Isaac Barrow]], is difficult to determine because of the priority dispute between them.<ref>See, for example:
* {{cite book|last=Mahoney |first=Michael S. |year=1990 |chapter=Barrow's mathematics: Between ancients and moderns |title=Before Newton |editor-first=M. |editor-last=Feingold |pages=179–249 |publisher=Cambridge University Press |isbn=978-0-521-06385-2}}
* {{Cite journal |first=M. |last=Feingold |date=June 1993 |title=Newton, Leibniz, and Barrow Too: An Attempt at a Reinterpretation |journal=[[Isis (journal)|Isis]] |language=en |volume=84 |issue=2 |pages=310–338 |doi=10.1086/356464 |bibcode=1993Isis...84..310F |s2cid=144019197 |issn=0021-1753}}
* {{cite book|first=Siegmund |last=Probst |chapter=Leibniz as Reader and Second Inventor: The Cases of Barrow and Mengoli |title=G.W. Leibniz, Interrelations Between Mathematics and Philosophy|editor-first1=Norma B. |editor-last1=Goethe |editor-first2=Philip |editor-last2=Beeley |editor-first3=David |editor-last3=Rabouin |publisher=Springer |isbn=978-9-401-79663-7 |pages=111–134 |year=2015 |series=Archimedes: New Studies in the History and Philosophy of Science and Technology |volume=41}}</ref>) The fundamental theorem provides an algebraic method of computing many definite integrals—without performing limit processes—by finding formulae for [[antiderivative]]s. It is also a prototype solution of a [[differential equation]]. Differential equations relate an unknown function to its derivatives and are ubiquitous in the sciences.<ref>{{Cite book |last1=Herman |first1=Edwin |url=https://openstax.org/details/books/calculus-volume-2 |title=Calculus. Volume 2 |last2=Strang |first2=Gilbert |date=2017 |publisher=OpenStax |isbn=978-1-5066-9807-6 |location=Houston |oclc=1127050110 |display-authors=etal |access-date=26 July 2022 |archive-date=26 July 2022 |archive-url=https://web.archive.org/web/20220726140351/https://openstax.org/details/books/calculus-volume-2 |url-status=live }}</ref>{{Rp|pages=351–352}}