Chapter 14.1, Problem 39ES

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.

Suppose that the temperature in degrees Celsius at a point $\left(x,y\right)$ on a flat metal plate is $T\left(x,y\right)=10-8x^2-2y^2,$ where $x$ and $y$ are in meters. Find the average temperature of the rectangular portion of the plate for which $0\le x\le 1\text{and}\text{0}\le y\le 2.$