Answer
$$\int^{\beta}_{\alpha} \dfrac{1}{2}r^2 d\theta $$
Work Step by Step
Our aim is to integrate the integral and find the area using polar coordinates as follows:
$ Area= \int^{\beta}_{\alpha} \int^{f(\theta)}_{0} r \space dr \space d\theta =\int^{\beta}_{\alpha} [\dfrac{r^2}{2}]^{f(\theta)}_0 d\theta $
or, $=\dfrac{1}{2}\int^{\beta}_{\alpha} f^2(\theta) d\theta $
Plug $ r=f(\theta)$, so we get:
$ Area=\int^{\beta}_{\alpha} \dfrac{1}{2}r^2 d\theta $
