Chapter 15.R, Problem 58E

The Mean Value Theorem for double integrals says that if f is a continuous function on a plane region D that is of type I or II, then there exists a point $\left(x_0,y_0\right)$ in D such that

${\displaystyle \underset{D}{\iint }f\left(x,y\right)}dA=f\left(x_0,y_0\right)A\left(D\right)$

Use the Extreme Value Theorem (14.7.8) and Property 15.2.11 of integrals to prove this theorem. (Use the proof of the single-variable version in Section 5.5 as a guide.)