Chapter 5.3, Problem 174E

Let f be a continuous function that can be expressed in polar coordinates as a function of 0 only: that is, $f\left(x,y\right)=h\left(\theta \right)$. where $\left(x,y\right)\in D=\left\{r,\theta \right\}|R_1\le r\le R_2,\theta \le {\theta }_2$with $0\le R_1<R_2$, and $0\le {\theta }_1<{\theta }_2\le 2\pi$ . Show that ${\displaystyle \underset{D}{\iint }f\left(x,y\right)dA}=\frac{1}{2}\left(R_2^2-R_1^2\right)H\left({\theta }_2\right)-H\left({\theta }_1\right)$. where H is an antiderivative of h.