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.