Chapter 14.5, Problem 34ES

The average value or mean value of a continuous function $f\left(x,y,z\right)$ over a solid G is defined as

$f_{\text{ave}}=\frac{1}{V\left(G\right)}{\displaystyle \underset{G}{\iiint }f\left(x,y,z\right)dV}$

Where $V\left(G\right)$ is the volume of the solid G (compare to the definition preceding Exercise 61 of Section 14.2). Use this definition in these exercises.

Find the average value of $f\left(x,y,z\right)=xyz$ over the spherical region $x^2+y^2+z^2\le 1.$