The graphs of the polar curves $r = 3$ and $r = 3 - 2 \sin ( 2 \theta )$ are shown in the figure above for $0 \leq \theta \leq \pi$. (a) Let $R$ be the shaded region that is inside the graph of $r = 3$ and inside the graph of $r = 3 - 2 \sin ( 2 \theta )$. Find the area of $R$. (b) For the curve $r = 3 - 2 \sin ( 2 \theta )$, find the value of $\frac { d x } { d \theta }$ at $\theta = \frac { \pi } { 6 }$. (c) The distance between the two curves changes for $0 < \theta < \frac { \pi } { 2 }$. Find the rate at which the distance between the two curves is changing with respect to $\theta$ when $\theta = \frac { \pi } { 3 }$. (d) A particle is moving along the curve $r = 3 - 2 \sin ( 2 \theta )$ so that $\frac { d \theta } { d t } = 3$ for all times $t \geq 0$. Find the value of $\frac { d r } { d t }$ at $\theta = \frac { \pi } { 6 }$.
The graphs of the polar curves $r = 3$ and $r = 3 - 2 \sin ( 2 \theta )$ are shown in the figure above for $0 \leq \theta \leq \pi$.\\
(a) Let $R$ be the shaded region that is inside the graph of $r = 3$ and inside the graph of $r = 3 - 2 \sin ( 2 \theta )$. Find the area of $R$.\\
(b) For the curve $r = 3 - 2 \sin ( 2 \theta )$, find the value of $\frac { d x } { d \theta }$ at $\theta = \frac { \pi } { 6 }$.\\
(c) The distance between the two curves changes for $0 < \theta < \frac { \pi } { 2 }$. Find the rate at which the distance between the two curves is changing with respect to $\theta$ when $\theta = \frac { \pi } { 3 }$.\\
(d) A particle is moving along the curve $r = 3 - 2 \sin ( 2 \theta )$ so that $\frac { d \theta } { d t } = 3$ for all times $t \geq 0$. Find the value of $\frac { d r } { d t }$ at $\theta = \frac { \pi } { 6 }$.