Answer
$$R = \left\{ {\left( {r,\theta } \right):4 \leqslant r \leqslant 8,{\text{ 0}} \leqslant \theta \leqslant \frac{\pi }{2}} \right\}$$
Work Step by Step
$$\eqalign{
& {\text{The represented region }}R{\text{ is a quarter of all points between }} \cr
& {\text{concentric circles of radius 4 and 8}}{\text{. It can be described}} \cr
& {\text{in polar coordinates as}} \cr
& R = \left\{ {\left( {r,\theta } \right):4 \leqslant r \leqslant 8,{\text{ 0}} \leqslant \theta \leqslant \frac{\pi }{2}} \right\} \cr} $$
