The graphs of the polar curves $r = 3$ and $r = 4 - 2 \sin \theta$ are shown in the figure. The curves intersect when $\theta = \frac { \pi } { 6 }$ and $\theta = \frac { 5 \pi } { 6 }$.
(a) Let $S$ be the shaded region that is inside the graph of $r = 3$ and also inside the graph of $r = 4 - 2 \sin \theta$. Find the area of $S$.
(b) A particle moves along the polar curve $r = 4 - 2 \sin \theta$ so that at time $t$ seconds, $\theta = t ^ { 2 }$. Find the time $t$ in the interval $1 \leq t \leq 2$ for which the $x$-coordinate of the particle's position is $-1$.
(c) For the particle described in part (b), find the position vector in terms of $t$. Find the velocity vector at time $t = 1.5$.