Chapter 15.6, Problem 54E

53–54 The average value of a function $f\left(x,y,z\right)$  over a solid region E is defined to be

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

where $V\left(E\right)$ is the volume of E. For instance, if $\rho$ is a density function, then ${\rho }_{ave}$ is the average density of E.

Find the average height of the points in the solid hemisphere $x^2+y^2+z^2\le 1,z\ge 0$.