## Thomas' Calculus 13th Edition

$\int^{\beta}_{\alpha} \frac{1}{2}r^2 d\theta$
We calculate the area using polar coordinates as follows: $A= \int^{\beta}_{\alpha} \int^{f(\theta)}_{0} rdrd\theta$ =$\int^{\beta}_{\alpha} [\frac{r^2}{2}]^{f(\theta)}_0 d\theta$ =$\frac{1}{2}\int^{\beta}_{\alpha} f^2(\theta) d\theta$ Substitute $r=f(\theta)$: =$\int^{\beta}_{\alpha} \frac{1}{2}r^2 d\theta$