Chapter 15.6, Problem 55E

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.

53. Find the average value of the function f(x, y, z) = xyz over the cube with side length L that lies in the first octant with one vertex at the origin and edges parallel to the coordinate axes.