The polar curve $r$ is given by $r ( \theta ) = 3 \theta + \sin \theta$, where $0 \leq \theta \leq 2 \pi$. (a) Find the area in the second quadrant enclosed by the coordinate axes and the graph of $r$. (b) For $\frac { \pi } { 2 } \leq \theta \leq \pi$, there is one point $P$ on the polar curve $r$ with $x$-coordinate - 3 . Find the angle $\theta$ that corresponds to point $P$. Find the $y$-coordinate of point $P$. Show the work that leads to your answers. (c) A particle is traveling along the polar curve $r$ so that its position at time $t$ is $( x ( t ) , y ( t ) )$ and such that $\frac { d \theta } { d t } = 2$. Find $\frac { d y } { d t }$ at the instant that $\theta = \frac { 2 \pi } { 3 }$, and interpret the meaning of your answer in the context of the problem.
: x \text {-coordinate }
The polar curve $r$ is given by $r ( \theta ) = 3 \theta + \sin \theta$, where $0 \leq \theta \leq 2 \pi$.
(a) Find the area in the second quadrant enclosed by the coordinate axes and the graph of $r$.
(b) For $\frac { \pi } { 2 } \leq \theta \leq \pi$, there is one point $P$ on the polar curve $r$ with $x$-coordinate - 3 . Find the angle $\theta$ that corresponds to point $P$. Find the $y$-coordinate of point $P$. Show the work that leads to your answers.
(c) A particle is traveling along the polar curve $r$ so that its position at time $t$ is $( x ( t ) , y ( t ) )$ and such that $\frac { d \theta } { d t } = 2$. Find $\frac { d y } { d t }$ at the instant that $\theta = \frac { 2 \pi } { 3 }$, and interpret the meaning of your answer in the context of the problem.