Answer
Use polar coordinates and write $\iint_Rf(x,y)dA=\int_0^4\int_0^{3\pi/2}f(r\cos\theta,r\sin\theta)rd\theta dr$.
Work Step by Step
The region $R$ can be easily written as the set of points in polar coordinates, that is $R=\{(r,\theta)|0\leq r\leq 4,0\leq \theta\leq \frac{3\pi}{2}\}$.
It means that we decide to use polar coordinates and the integral can be expressed $\iint_Rf(x,y)dA=\int_0^4\int_0^{3\pi/2}f(r\cos\theta,r\sin\theta)rd\theta dr$
