The curve above is drawn in the $x y$-plane and is described by the equation in polar coordinates $r = \theta + \sin ( 2 \theta )$ for $0 \leq \theta \leq \pi$, where $r$ is measured in meters and $\theta$ is measured in radians. The derivative of $r$ with respect to $\theta$ is given by $\frac { d r } { d \theta } = 1 + 2 \cos ( 2 \theta )$. (a) Find the area bounded by the curve and the $x$-axis. (b) Find the angle $\theta$ that corresponds to the point on the curve with $x$-coordinate $-2$. (c) For $\frac { \pi } { 3 } < \theta < \frac { 2 \pi } { 3 } , \frac { d r } { d \theta }$ is negative. What does this fact say about $r$ ? What does this fact say about the curve? (d) Find the value of $\theta$ in the interval $0 \leq \theta \leq \frac { \pi } { 2 }$ that corresponds to the point on the curve in the first quadrant with greatest distance from the origin. Justify your answer.
The curve above is drawn in the $x y$-plane and is described by the equation in polar coordinates $r = \theta + \sin ( 2 \theta )$ for $0 \leq \theta \leq \pi$, where $r$ is measured in meters and $\theta$ is measured in radians. The derivative of $r$ with respect to $\theta$ is given by $\frac { d r } { d \theta } = 1 + 2 \cos ( 2 \theta )$.\\
(a) Find the area bounded by the curve and the $x$-axis.\\
(b) Find the angle $\theta$ that corresponds to the point on the curve with $x$-coordinate $-2$.\\
(c) For $\frac { \pi } { 3 } < \theta < \frac { 2 \pi } { 3 } , \frac { d r } { d \theta }$ is negative. What does this fact say about $r$ ? What does this fact say about the curve?\\
(d) Find the value of $\theta$ in the interval $0 \leq \theta \leq \frac { \pi } { 2 }$ that corresponds to the point on the curve in the first quadrant with greatest distance from the origin. Justify your answer.