Chapter 15.9, Problem 21E

(a) Evaluate ${\displaystyle {\iiint }_{E}dV}$, where $E$ is the solid enclosed by the ellipsoid $x^2/a^2+y^2/b^2+z^2/c^2=1$. Use the transformation $x=au,y=bv,z=cw$.

(b) The earth is not a perfect sphere; rotation has resulted in flattening at the poles. So the shape can be approximated by an ellipsoid with $a=b=6378\text{ km}$ and $c=6356\text{ km}$. Use part (a) to estimate the volume of the earth.

(c) If the solid of part (a) has constant density k, find its moment of inertia about the z-axis.