Chapter 5.3, Problem 170E

A radial function f is a function whose value at each point depends only on the distance between that point and the origin of the system of coordinates: that is,

$f\left(x,y\right)=g\left(r\right)$. where $r=\sqrt{x^2+y^2}$. Show that if f is a continuous radial function, then ${\displaystyle \underset{D}{\iint }f\left(x,y\right)dA=\left({\theta }_2-{\theta }_1\right)\left[G\left(R_2\right)-G\left(R_1\right)\right]}$. where

G’(r) = rg(r) and $\left(x,y\right)\in D=\left\{\left(r,\theta \right)|R_1\le r\le R_2,0\le 2\pi \right\}$ , with $0\le R_1<R_2$and $0\le {\theta }_1<{\theta }_2\le 2\pi$.