The calculation is derived from the ancient Greek geometry. More information is housed here: 4Moms. Democritus calculated the volume of pyramids and cones, it is believed that you considering them formed by an infinite number of sections of infinitesimal thickness (infinitely small), and Eudoxus and Archimedes used the method of exhaustion to find the area of a circle with the accuracy required by the use of polygons inscribed. However, the difficulties to work with irrational numbers and the paradoxes of Zeno of Eleaimpidieron formulate a systematic theory of the calculation. In the 17TH century, Francesco B. Cavalieri yEvangelista Torricelli expanded the use of infinitesimals, and Descartes and Pierre de Fermat used the algebra to find the area and the tangents (integration and differentiation in modern terms).
Fermat and Isaac Barrow had the certainty that both calculations were related, although they were Isaac Newton (c. 1660) and Gottfried W. Leibniz (circa 1670) who showed that they are inverses, which is known as the fundamental theorem of calculus. The discovery of Newton, from his theory of gravity, was prior to Leibniz’s, but the delay in their publication still causes disputes about who was first. However, he eventually adopted the Leibniz notation. In the 18th century it increased considerably the number of applications of the calculation, but the imprecise use of the infinite and infinitesimal quantities, as well as geometric intuition, caused still confusion and controversy about its foundations. One of his most notable critics was the Irish philosopher George Berkeley. In the 19th century mathematical analysts replaced such vagueness by solid foundations based on finite quantities: Bernhard Bolzano and Augustin Louis Cauchy defined with precision the limits and derivatives; Cauchy and Bernhard Riemann did the same with the integrals, and Julius Dedekind and Karl Weierstrass with real numbers. For example, it was learned that differentiable functions are continuous and continuous functions are integrable, although the reciprocals are false.