Summary of Area and Arc Length in Polar Coordinates

The area of a region in polar coordinates defined by the equation $r=f\left(\theta \right)$ with $\alpha \le \theta \le \beta$ is given by the integral $A=\frac{1}{2}{{\displaystyle\int }_{\alpha }^{\beta }\left[f\left(\theta \right)\right]}^{2}d\theta$ .
To find the area between two curves in the polar coordinate system, first find the points of intersection, then subtract the corresponding areas.
The arc length of a polar curve defined by the equation $r=f\left(\theta \right)$ with $\alpha \le \theta \le \beta$ is given by the integral $L={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{\left[f\left(\theta \right)\right]}^{2}+{\left[{f}^{\prime }\left(\theta \right)\right]}^{2}}d\theta ={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{r}^{2}+{\left(\frac{dr}{d\theta }\right)}^{2}}d\theta$ .

Key Equations

Area of a region bounded by a polar curve
$A=\frac{1}{2}{\displaystyle\int }_{\alpha }^{\beta }{\left[f\left(\theta \right)\right]}^{2}d\theta =\frac{1}{2}{\displaystyle\int }_{\alpha }^{\beta }{r}^{2}d\theta$
Arc length of a polar curve
$L={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{\left[f\left(\theta \right)\right]}^{2}+{\left[{f}^{\prime }\left(\theta \right)\right]}^{2}}d\theta ={\displaystyle\int }_{\alpha }^{\beta }\sqrt{{r}^{2}+{\left(\frac{dr}{d\theta }\right)}^{2}}d\theta$