The Fundamental Theorem of Calculus and Accumulation Functions

Picture an accumulation function g(x), the integral from some constant a to x of f(t) dt. It measures running area under f, swept out from a up to wherever x currently sits. As x slides right by a tiny amount dx, the area gains one thin sliver, a rectangle roughly f(x) tall and dx wide. So the extra area divided by dx is about f(x), and that approximation gets exact as dx shrinks to zero. That is the whole theorem: g'(x) = f(x). The rate the area grows equals the height of the curve at the edge doing the growing, nothing about the curve's past behavior matters for the slope right now.

This flips integration and differentiation into inverse operations. Finding an antiderivative is just finding some function whose derivative matches f, and accumulation functions hand you one automatically. When the upper limit is not plain x but some function of x, like x², the chain rule tacks on: differentiate the outer accumulation, substitute the upper limit into f, then multiply by the derivative of that limit. This is why g(x) = integral from 2 to x of √(t³+1) dt gives g'(x) = √(x³+1) directly, while F(x) = integral from 0 to x² of cos(t) dt gives F'(x) = cos(x²) times 2x.

This sits right after you learn what a definite integral means as a limit of Riemann sums, and right before you start building a toolkit of antiderivative techniques, since the theorem is the reason antiderivatives compute area in the first place.