Curve $C$ is defined by the polar equation $r ( \theta ) = 2 \sin ^ { 2 } \theta$ for $0 \leq \theta \leq \pi$. Curve $C$ and the semicircle $r = \frac { 1 } { 2 }$ for $0 \leq \theta \leq \pi$ are shown in the $x y$-plane.
(Note: Your calculator should be in radian mode.)
A. Find the rate of change of $r$ with respect to $\theta$ at the point on curve $C$ where $\theta = 1.3$. Show the setup for your calculations.
B. Find the area of the region that lies inside curve $C$ but outside the graph of the polar equation $r = \frac { 1 } { 2 }$. Show the setup for your calculations.
C. It can be shown that $\frac { d x } { d \theta } = 4 \sin \theta \cos ^ { 2 } \theta - 2 \sin ^ { 3 } \theta$ for curve $C$. For $0 \leq \theta \leq \frac { \pi } { 2 }$, find the value of $\theta$ that corresponds to the point on curve $C$ that is farthest from the $y$-axis. Justify your answer.
D. A particle travels along curve $C$ so that $\frac { d \theta } { d t } = 15$ for all times $t$. Find the rate at which the particle's distance from the origin changes with respect to time when the particle is at the point where $\theta = 1.3$. Show the setup for your calculations.