Chapter 15.9, Problem 21E

(a) Evaluate ∫∫∫_E dV, where E is the solid enclosed by the ellipsoid x₂/a₂ + y₂/b² + z²/c² = 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 km and c = 6356 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.