(874, #47) Find the average of the function f(x,yz) = x y z 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 axis.

Solution: The integral of xyz on the cube is

Since the bounds of integration are constants this triple integral factors into a product of three simple integrals:

Finally, since the volume of the cube is L³ the average value is L³/8.

by: gm