mean value theorem for integrals

The average height of a continuous curve over an interval is the integral divided by the interval's length. The mean value theorem for integrals says that a continuous function actually attains this average at some point: there is a place where the curve's height equals its own average, so a rectangle of that height and the interval's width has exactly the same area as the region under the curve.

Precisely, if f is continuous on [a, b], then there exists a point c in [a, b] such that the integral of f from a to b equals f(c)(b - a). Equivalently, f(c) = (1/(b - a)) times the integral of f over [a, b], which is the average value of f. The proof combines the extreme value theorem (f attains a min m and max M) with the intermediate value theorem applied to the average, which lies between m and M.

Continuity is essential here, unlike in the Riemann criterion. A merely integrable function need not hit its average — a step function jumping between 0 and 1 has average 1/2 but never takes the value 1/2. There is also a weighted version: if g >= 0 is integrable, the integral of f g equals f(c) times the integral of g for some c, a tool used in remainder estimates for Taylor's theorem.