A polar arch: area, distance from the y-axis, motion toward the origin and a tangent slope

The question

Name: ______________________   Date: ____________   A polar arch: area, distance from the y-axis, motion toward the origin and a tangent slope (9 points)

Graph

The curve $r = 3\sin\theta + \cos(2\theta)$

The curve shown is the graph of the polar equation $r = 3\sin\theta + \cos(2\theta)$ for $0 \le \theta \le \pi$. It starts at the point $(1, 0)$ and ends at $(-1, 0)$. Region $R$ is bounded by the curve and the $x$-axis. A graphing calculator is required, in radian mode.

1. (a) Find the area of region $R$. [3 points]

2. (b) For $0 \le \theta \le \frac{\pi}{2}$, find the greatest distance from a point on the curve to the $y$-axis. Show the work that leads to your answer. [2 points]

3. (c) Is the point on the curve moving toward the origin or away from the origin as $\theta$ increases through $\theta = \frac{5\pi}{6}$? Give a reason for your answer. [2 points]

4. (d) Find the slope of the line tangent to the curve at the point where $\theta = \frac{\pi}{3}$. [2 points]