Integral Calculus, Mean Value Theorem

Mean Value Theorem

A continuous function attains its average somewhere. But first we must define average.

The average of a function f over an interval [a,b] is the integral of f divided by b-a. More generally, the average of a function over a multidimensional region is the integral of f divided by the volume of the region.

The one dimensional version of this theorem can be proved by joining the fundamental theorem of calculus with the mean value theorem. However, I will present a proof that is both simpler and more general. It works in higher dimensions. For instance, there is a point somewhere in the sun that attains the average temperature of the sun.

Assume f is continuous and bounded over a finite region. If the region is closed, then continuous implies bounded. Let l be the lower bound and u be the upper bound. Let v be the volume of the region. Let s be the integral of f over the region. Note that l×v ≤ s ≤ u×v. Since f is continuous it attains every value between l and u, including s/v, which happens to be the average of f.