Summary of Double Integrals in Polar Coordinates

To apply a double integral to a situation with circular symmetry, it is often convenient to use a double integral in polar coordinates. We can apply these double integrals over a polar rectangular region or a general polar region, using an iterated integral similar to those used with rectangular double integrals.
The area $dA$ in polar coordinates becomes $rdrd\theta$ .
Use $x=r\cos\theta$ , $y=r\sin\theta$ , and $dA=rdrd\theta$ to convert an integral in rectangular coordinates to an integral in polar coordinates.
Use $r^{2}=x^{2}+y^{2}$ and $\theta=\tan^{-1}\left(\frac{y}{x}\right)$ to convert an integral in polar coordinates to an integral in rectangular coordinates, if needed.
To find the volume in polar coordinates bounded above by a surface $z=f(r,\theta)$ over a region on the $xy$ -plane, use a double integral in polar coordinates.

Key Equations

Double integral over a polar rectangular region
$\underset{R}{\displaystyle\iint} f(r,\theta)dA = \underset{m,n\to\infty}{\lim}\displaystyle\sum_{i=1}^{m}\displaystyle\sum_{j=1}^{n}f({r_{ij}}^{\ast},{\theta_{ij}}^{\ast})\Delta{A} =\underset{m,n\to\infty}{\lim}\displaystyle\sum_{i=1}^{m}\displaystyle\sum_{j=1}^{n}f({r_{ij}}^{\ast},{\theta_{ij}}^{\ast}){r_{ij}}^{\ast}\Delta{r}\Delta{\theta}$
Double integral over a general polar region
$\underset{D}{\displaystyle\iint} f(r,\theta)r dr d\theta=\displaystyle\int_{\theta=\alpha}^{\theta=\beta} \displaystyle\int_{r=h_{1}(\theta)}^{r=h_{2}(\theta)}f(r,\theta)r dr d\theta$

polar rectangle: the region enclosed between the circles $r=a$ and $r=b$ and the angles $\theta = \alpha$ and $\theta = \beta$ ; it is described as ${\bf{R}}=\{(r,{\theta}) | a{\leq}r{\leq}b, {\alpha}{\leq}{\theta}{\leq}{\beta}\}$