Chapter 15, Problem 58RE

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 (x₀, y₀) in D such that ${\displaystyle \underset{D}{\iint }f(x,y)dA=f(x_0,y_0)A(D)}$ Use the Extreme Value Theorem (14.7.8) and Properly 15.2.11 of integrals to prove this theorem. (Use the proof of the single-variable version in Section 6.5 as a guide.)