Answer
$$R = \left\{ {\left( {r,\theta } \right):0 \leqslant r \leqslant 2\sin \theta ,{\text{ 0}} \leqslant \theta \leqslant \pi } \right\}$$
Work Step by Step
$$\eqalign{
& {\text{The represented region }}R{\text{ is a circle of radius 2 centered at}} \cr
& \left( {0,2} \right){\text{. }} \cr
& {x^2} + {y^2} = {\left( 2 \right)^2} \cr
& {x^2} + {\left( {y - 2} \right)^2} = 4 \cr
& {x^2} + {y^2} - 2y + 4 = 4 \cr
& {x^2} + {y^2} - 2y = 0 \cr
& x = r\cos \theta ,{\text{ }}y = r\sin \theta \cr
& {r^2}{\cos ^2}\theta + {r^2}{\sin ^2}\theta - 2r\sin \theta = 0 \cr
& {r^2}\left( {{{\cos }^2}\theta + {{\sin }^2}\theta } \right) = 2r\sin \theta \cr
& r = 2\sin \theta \cr
& {\text{It can be described in polar coordinates as:}} \cr
& R = \left\{ {\left( {r,\theta } \right):0 \leqslant r \leqslant 2\sin \theta ,{\text{ 0}} \leqslant \theta \leqslant \pi } \right\} \cr} $$
