Chapter 15.6, Problem 53E

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 value of the function $f\left(x,y,z\right)=xyz$  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.