One theorem proved differentiation and integration are inverses
The fundamental theorem of calculus links finding slopes with finding areas, showing that differentiation and integration roughly undo each other. Greek geometers could already work out areas using infinitesimals centuries earlier, but nobody had realized the two operations were connected until the theorem was proved.
The theorem comes in two parts. The first says that if f is continuous, integrating it from a fixed point up to a moving endpoint produces an antiderivative, so the rate at which that accumulated area grows equals f at the endpoint. The second says the integral over a fixed interval [a, b] equals F(b) minus F(a) for any antiderivative F, which lets symbolic integration replace laborious numerical approximation. That second part is also called the Newton–Leibniz theorem, and naming conventions are not fully standardized.
The intuition for the first part uses a thin strip under the curve. Its area between x and x + h is A(x + h) − A(x), and it is also close to a rectangle of height f(x) and width h. Dividing by h and letting h shrink to zero turns the approximation into an equality, so the derivative of the area function is the original function.
A car trip illustrates the second part. A passenger who can see only the speedometer can multiply the speed by each one-second interval and add the results to estimate the distance covered. As the intervals shrink the sum becomes an integral, and integrating velocity, the derivative of position, gives the net change in position; adding the starting point gives the location on the highway.
The ingredients were old: Greek mathematicians computed areas with infinitesimals, and in the fourteenth century the Oxford Calculators studied continuity and motion. James Gregory (1638–1675) published the first proof of a rudimentary, strongly geometric form. Isaac Barrow proved a more general version, his pupil Isaac Newton completed the surrounding theory, and Gottfried Leibniz organized it into a calculus of infinitesimals with the notation still in use.
Source: Fundamental theorem of calculus
