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 [latex]dA[/latex] in polar coordinates becomes [latex]rdrd\theta[/latex].
Use [latex]x=r\cos\theta[/latex], [latex]y=r\sin\theta[/latex], and [latex]dA=rdrd\theta[/latex] to convert an integral in rectangular coordinates to an integral in polar coordinates.
Use [latex]r^{2}=x^{2}+y^{2}[/latex] and [latex]\theta=\tan^{-1}\left(\frac{y}{x}\right)[/latex] 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 [latex]z=f(r,\theta)[/latex] over a region on the [latex]xy[/latex]-plane, use a double integral in polar coordinates.

Key Equations

Double integral over a polar rectangular region
[latex]\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}[/latex]
Double integral over a general polar region
[latex]\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[/latex]

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