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Section 15.8: Triple Integration in Spherical Coordinates

Exercise 49

  1. Use cylindrical coordinates to show that the volume of the solid bounded above by the sphere \(r^2+z^2=a^2\) and below by the cone \(z=r\cot\phi_0\) (or \(\phi=\phi_0\)), where \(0 < \phi_0 < \pi/2\), is \[ V = \frac{2\pi a^3}{3}(1-\cos\phi_0) \]
  2. Deduce a that the volume of the spherical wedge given by \(\rho_1 \le \rho \le \rho_2, \theta_1 \le \theta \le \theta_2, \phi_1 \le \phi \le \phi_2\) is \[ \Delta V = \frac{\rho_2^3 - \rho_1^3}{3}(\cos\phi_1 - \cos\phi_2)(\theta_2 - \theta_1). \]
  3. Use the Mean Value Theorem to show that the volume in part (b) can be written as \[ \Delta V = \tilde{\rho}^2 \sin\tilde{\phi} \, \Delta\rho \, \Delta\theta \, \Delta\phi \] where \(\tilde{\rho}\) lies between \(\rho_1\) and \(\rho_2\), \(\tilde{\phi}\) lies between \(\phi_1\) and \(\phi_2\), \(\Delta\rho = \rho_2 - \rho_1\), \(\Delta\theta = \theta_2 - \theta_1\), and \(\Delta\phi = \phi_2 - \phi_1\).