fundamental theorem of calculus

Differentiation measures instantaneous rate; integration accumulates total change. The fundamental theorem says these two operations are inverse to each other: accumulating a rate and then differentiating gives the rate back, and differentiating a quantity and then accumulating its rate returns the net change. This is why you can compute hard areas just by reversing a derivative.

There are two parts. The first part says that if f is continuous on [a, b] and F(x) is the integral of f from a to x, then F is differentiable and F'(x) = f(x): the area-so-far function differentiates back to the integrand. The second part (the evaluation rule) says that if g is any antiderivative of a Riemann integrable f, with g' = f on [a, b], then the integral of f from a to b equals g(b) - g(a).

The hypotheses matter and are often blurred. Part one needs continuity of f at the point in question to conclude differentiability there; at a jump discontinuity the area function is continuous but not differentiable. Part two does not require f to be continuous, only Riemann integrable with an everywhere antiderivative — but beware that not every Riemann integrable function has an antiderivative, and not every function with an antiderivative is Riemann integrable, so the two parts are genuinely separate statements.