fundamental theorem of calculus

The fundamental theorem of calculus is the discovery that the two great operations of calculus — finding a rate (differentiation) and finding a total (integration) — are exact opposites of each other, like multiplication and division. It says that adding up infinitely many tiny changes gets you back the net change, and that the running total of a rate is itself growing at that very rate. This is the bridge that turns calculus from two separate subjects into one.

It has two parts. Part 1: if you define a function by an accumulating integral, A(x) = integral from a to x f(t) dt, then its derivative is just the integrand: A'(x) = f(x). Differentiating the running total hands back the rate. Part 2 (the evaluation part): if F is any antiderivative of f, then integral from a to b f(x) dx = F(b) - F(a). You compute a total by undoing a derivative and subtracting the two endpoint values.

Why this is revolutionary: the definite integral was defined as a limit of Riemann sums — an infinite process that looks impossible to carry out by hand. Part 2 says you never have to. Instead of summing infinitely many slivers, find one antiderivative and do a single subtraction. Centuries of separate work on tangent lines and on areas turned out to be two views of the same machine; that unification, made systematic by Newton and Leibniz in the late 1600s, is what launched modern science and engineering.