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Section 15.9: Change of Variables in Multiple Integrals

Exercise 21

  1. Evaluate \(\iiint_E dV\), where E is the solid enclosed by the ellipsoid \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\). Use the transformation \(x=au, y=bv, z=cw\).
  2. 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.
  3. If the solid of part (a) has constant density k, find its moment of inertia about the z-axis.