Section 15.6: Triple Integration

The average value of a function \(f(x, y, z)\) over a solid region E is defined to be \(f_{ave} = \frac{1}{V(E)} \iiint_E f(x, y, z) dV\) where \(V(E)\) 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 \(f(x, y, z) = xyz\) 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.