5. Polar Coordinates

Integral Over Polar Rectangle

Theorem

If the domain equals

$D=\{(r,\theta )|0\le a\le r\le b,\alpha \le \theta \le \beta ,0\le \beta -\alpha \le 2\pi \}$

in polar coordinates, then

${\iint }_{D}f(x,y)dA={\int }_{\alpha }^{\beta }{\int }_a^{b}f(r\cos (\theta ),r\sin (\theta ))rdrd\theta$

where

$dA=rdrd\theta$

$x=r\cos (\theta )$

$y=r\sin (\theta )$

Integral Over More General Region

Theorem

If the domain equals

$D=\{(r,\theta )|0\le h_1(\theta )\le r\le h_2(\theta ),\alpha \le \theta \le \beta ,0\le \beta -\alpha \le 2\pi \}$

in polar coordinates, then

${\iint }_{D}f(x,y)dA={\int }_{\alpha }^{\beta }{\int }_{h_1(\theta )}^{h_2(\theta )}f(r\cos (\theta ),r\sin (\theta ))rdrd\theta$