Chapter 14.1, Problem 40ES

The average value or mean value of a continuous function $f\left(x,y\right)$ $R=\left[a,b\right]\times \left[c,d\right]$ is defined as $f_{\text{ave}}=\frac{1}{A\left(R\right)}{\displaystyle \underset{R}{\iint }f\left(x,y\right)dA}$

Where $A\left(R\right)=\left(b-a\right)\left(d-c\right)$ is the area of the rectangle $R$ (compare to Definition 5.8.1). Use this definition in these exercises.

Show that if $f\left(x,y\right)$ is constant on the rectangle $R=\left[a,b\right]\times \left[c,d\right],\text{say}f\left(x,y\right)=k,\text{then}{f}_{\text{ave}}=k\text{over}R.$