## 31.6 Doing the Integrals

Once you have established limits of integration on your variables and have an integrand and the right set of differentials, you are ready to start integrating. You are then on your own.

Exercise 31.8 Determine the moments of inertia of bodies with the shape of the cone and ellipsoid above about the z-axis. Assume that the mass densities are constant throughout the bodies. (The moment of inertia of a body is the integral of its density multiplied by the square of the distance of the volume element to the z-axis. However, it is always expressed as a function not of the density, but of the mass of the body. You must therefore take the ratio of the integral determining the moment of inertia, and that determining the mass to get the relation between the two.)