15.8 Videos Guide

15.8a

• Rectangular-spherical conversions

  • $x=\rho\cos\theta\sin\phi$

  • $y=\rho\sin\theta\sin\phi$

  • $z=\rho\cos\phi$

  • $x^{2}+y^{2}+z^{2}=\rho^{2}$

15.8b

Exercise:

• Sketch the solid described by the given inequalities.
  $1\leq\rho\leq2$, $\frac{\pi}{2}\leq\phi\leq\pi$

• The triple integral in spherical coordinates

  • $\iiint^{\ }_{E}{f(x,\ y,\ z)\ dV}$

$$=\int^{d}_{c}{\int^{\beta}_{\alpha}{\int^{g_{2}(\theta,\ \phi)}_{g_{1}(\theta,\ \phi)}{f(}}\rho\cos\theta\sin\phi,\rho\sin\theta\sin\phi,\ \rho\cos\phi)\ \rho^{2}\sin\phi\ d\rho\ d\theta\ d\phi}$$

• Note that $dV=\rho^{2}\sin\phi\ d\rho\ d\theta\ d\phi$

Exercises:

15.8c

• Sketch the solid whose volume is given by the integral and evaluate the integral.
  $$\int^{\frac{\pi}{4}}_{0}{\int^{2\pi}_{0}{\int^{\sec\phi}_{0}{\rho^{2}\sin\phi\ d\rho\ d\theta\ d\phi}}}$$

15.8d

• Use spherical coordinates to evaluate $\iiint^{\ }_{E}{y^{2}x^{2}\ dV}$, where $E$ lies above the cone
  $\phi=\frac{\pi}{3}$ and below the sphere $\rho=1$.

15.8e

• Evaluate the integral by changing to spherical coordinates.
  $$\int^{a}_{-a}{\int^{\sqrt{a^{2}-y^{2}}}_{-\sqrt{a^{2}-y^{2}}}{\int^{\sqrt{a^{2}-x^{2}-y^{2}}}_{-\sqrt{a^{2}-x^{2}-y^{2}}}{\left(x^{2}z+y^{2}z+z^{3}\right)\ dz\ dx\ dy}}}$$