Definite and Indefinite

Integration comes in two flavors. A definite integral computes a specific total between two points — the area under a curve from x = 1 to x = 5, say. You get a single number: the accumulated quantity over that interval.

An indefinite integral finds the general antiderivative — a family of functions whose derivative gives back the original. It carries a constant C because many functions share the same slope. Definite integrals pin down C by fixing the endpoints.

The distinction matters in practice. A physicist finding total work between two positions uses a definite integral. An engineer recovering velocity from acceleration uses an indefinite one.