15.3 Videos Guide

15.3a

• Rectangular-polar conversions

  • $x=r\cos\theta$

  • $y=r\sin\theta$

  • $x^{2}+y^{2}=r^{2}$

  • $\tan\theta=\frac{y}{x}$

• The double integral in polar coordinates

  • $\iint^{\ }_{R}{f(x,\ y)\ dA}=\int^{\beta}_{\alpha}{\int^{b}_{a}{f\left(r\cos\theta,\ r\sin\theta\right)\ rdrd\theta}}$

AND $\iint^{\ }_{D}{f(x,\ y)\ dA}=\int^{\beta}_{\alpha}{\int^{h_{2}(\theta)}_{h_{1}(\theta)}{f\left(r\cos\theta,\ r\sin\theta\right)\ rdrd\theta}}$

• Note that $dV=rdrd\theta$

Exercises:

15.3b

• Sketch the region whose area is given by the integral and evaluate the integral.
  $$\int^{\pi}_{\frac{\pi}{2}}{\int^{2\sin\theta}_{0}{r\ dr\ d\theta}}$$

• Evaluate the integral by changing to polar coordinates.

$\iint^{\ }_{R}{(2x-y)\ dA}$, where $R$ is the region in the first quadrant enclosed by the circle
$x^{2}+y^{2}=4$ and the lines $x=0$ and $y=x$.

15.3c

• $\iint^{\ }_{D}{x\ dA}$, where $D$ is the region in the first quadrant that lies between the circles
  $x^{2}+y^{2}=4$ and $x^{2}+y^{2}=2x$

15.3d

• Use polar coordinates to find the volume of the solid below the cone $z=\sqrt{x^{2}+y^{2}}$ and above the ring $1\leq x^{2}+y^{2}\leq4$.

• Evaluate the iterated integral by converting to polar coordinates.
  $$\int^{a}_{0}{\int^{\sqrt{a^{2}-y^{2}}}_{-\sqrt{a^{2}-y^{2}}}{(2x+y)\ dx\ dy}}$$