Let $S$ be the region bounded by the graph of the polar curve $r ( \theta ) = 3 \sqrt { \theta } \sin \left( \theta ^ { 2 } \right)$ for $0 \leq \theta \leq \sqrt { \pi }$, as shown in the figure above. (a) Find the area of $S$. (b) What is the average distance from the origin to a point on the polar curve $r ( \theta ) = 3 \sqrt { \theta } \sin \left( \theta ^ { 2 } \right)$ for $0 \leq \theta \leq \sqrt { \pi }$? (c) There is a line through the origin with positive slope $m$ that divides the region $S$ into two regions with equal areas. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $m$. (d) For $k > 0$, let $A ( k )$ be the area of the portion of region $S$ that is also inside the circle $r = k \cos \theta$. Find $\lim _ { k \rightarrow \infty } A ( k )$.
Let $S$ be the region bounded by the graph of the polar curve $r ( \theta ) = 3 \sqrt { \theta } \sin \left( \theta ^ { 2 } \right)$ for $0 \leq \theta \leq \sqrt { \pi }$, as shown in the figure above.
(a) Find the area of $S$.
(b) What is the average distance from the origin to a point on the polar curve $r ( \theta ) = 3 \sqrt { \theta } \sin \left( \theta ^ { 2 } \right)$ for $0 \leq \theta \leq \sqrt { \pi }$?
(c) There is a line through the origin with positive slope $m$ that divides the region $S$ into two regions with equal areas. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $m$.
(d) For $k > 0$, let $A ( k )$ be the area of the portion of region $S$ that is also inside the circle $r = k \cos \theta$. Find $\lim _ { k \rightarrow \infty } A ( k )$.