15.7 Videos Guide

15.7a

• Rectangular-cylindrical conversions

  • $x=r\cos\theta$

  • $y=r\sin\theta$

  • $z=z$

  • $x^{2}+y^{2}=r^{2}$

  • $\tan\theta=\frac{y}{x}$

• The triple integral in cylindrical coordinates

  • $\iiint^{\ }_{E}{f(x,\ y,\ z)\ dV}=\int^{\beta}_{\alpha}{\int^{h_{2}(\theta)}_{h_{1}(\theta)}{\int^{u_{2}(r\cos\theta,\ r\sin\theta)}_{u_{1}(r\cos\theta,\ r\sin\theta)}{f\left(r\cos\theta,\ r\sin\theta\right)\ r\ dz\ dr\ d\theta}}}$

  • Note that $dV=r\ dz\ dr\ d\theta$

Exercises:

• Sketch the solid described by the given inequalities.
  $0\leq\theta\leq\frac{\pi}{2}$, $r\leq z\leq2$

• Use cylindrical coordinates to evaluate $\iiint^{\ }_{E}{z\ dV}$, where $E$ is enclosed by the paraboloid $z=x^{2}+y^{2}$ and the plane $z=4$.

15.7b

• Use cylindrical coordinates to evaluate $\iiint^{\ }_{E}{(x-y)\ dV}$, where $E$ is the solid that lies between the cylinders $x^{2}+y^{2}=1$ and $x^{2}+y^{2}=16$, above the $xy$-plane, and below the plane $z=y+4$.

15.7c

• Use cylindrical coordinates to find the volume of the solid that lies within both the cylinder $x^{2}+y^{2}=1$ and the sphere $x^{2}+y^{2}+z^{2}=4$.

• Evaluate the integral by changing to cylindrical coordinates.
  $$\int^{3}_{-3}{\int^{\sqrt{9-x^{2}}}_{0}{\int^{9-x^{2}-y^{2}}_{0}{\sqrt{x^{2}+y^{2}}\ dz\ dy\ dx}}}$$

15.7d

• Use cylindrical coordinates to find the volume of the solid that lies between the paraboloid $z=x^{2}+y^{2}$ and the sphere $x^{2}+y^{2}+z^{2}=2$.