The graphs of the polar curves $r = 4$ and $r = 3 + 2 \cos \theta$ are shown in the figure above. The curves intersect at $\theta = \frac { \pi } { 3 }$ and $\theta = \frac { 5 \pi } { 3 }$.
(a) Let $R$ be the shaded region that is inside the graph of $r = 4$ and also outside the graph of $r = 3 + 2 \cos \theta$, as shown in the figure above. Write an expression involving an integral for the area of $R$.
(b) Find the slope of the line tangent to the graph of $r = 3 + 2 \cos \theta$ at $\theta = \frac { \pi } { 2 }$.
(c) A particle moves along the portion of the curve $r = 3 + 2 \cos \theta$ for $0 < \theta < \frac { \pi } { 2 }$. The particle moves in such a way that the distance between the particle and the origin increases at a constant rate of 3 units per second. Find the rate at which the angle $\theta$ changes with respect to time at the instant when the position of the particle corresponds to $\theta = \frac { \pi } { 3 }$. Indicate units of measure.