definite integral

A definite integral is the total you accumulate as you sweep a changing quantity across an interval. If a function tells you the rate at which water flows into a tank at each instant, the definite integral over an hour tells you how much water arrived in that hour. It is a single number — the grand total — squeezed out of infinitely many infinitely small contributions.

Precisely, the definite integral of f from a to b, written integral from a to b f(x) dx, is defined as the LIMIT of Riemann sums as the slices become infinitely thin: integral from a to b f(x) dx = lim (n -> infinity) sum f(x_i) dx, with dx -> 0. It is not defined as 'the area under the curve' — that is only the picture in the simplest case. When f dips below the x-axis its contribution is counted as negative, so the integral measures net signed accumulation: area above the axis minus area below it. If f sits entirely above the axis on the interval, then the integral does equal the geometric area, which is why the area picture is so often taught first.

The result is a pure number that depends only on f and the endpoints a and b — the variable x is a 'dummy,' so integral from a to b f(x) dx and integral from a to b f(t) dt mean exactly the same thing. The miracle is that you almost never compute this limit directly: the Fundamental Theorem of Calculus lets you find an antiderivative F and simply subtract, F(b) - F(a). That shortcut turns a hopeless-looking infinite sum into a single subtraction.