## 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.