Chapter 15.6, Problem 54E

The average value of a function f (x, y, z) over a solid region E is defined to be $f_{ave}=\frac{1}{V(E)}{\displaystyle \underset{E}{\iiint }f\text{}(x,y,z)dV}$ where V(E) is the volume of E. For instance, if ρ is a density function, then ρ_ave is the average density of E.

54. Find the average height of the points in the solid hemisphere x² + y² + z² ≤ 1, z ≥ 0.