The figure shows the polar curves $r = f(\theta) = 1 + \sin\theta\cos(2\theta)$ and $r = g(\theta) = 2\cos\theta$ for $0 \leq \theta \leq \frac{\pi}{2}$. Let $R$ be the region in the first quadrant bounded by the curve $r = f(\theta)$ and the $x$-axis. Let $S$ be the region in the first quadrant bounded by the curve $r = f(\theta)$, the curve $r = g(\theta)$, and the $x$-axis.
(a) Find the area of $R$.
(b) The ray $\theta = k$, where $0 < k < \frac{\pi}{2}$, divides $S$ into two regions of equal area. Write, but do not solve, an equation involving one or more integrals whose solution gives the value of $k$.
(c) For each $\theta$, $0 \leq \theta \leq \frac{\pi}{2}$, let $w(\theta)$ be the distance between the points with polar coordinates $(f(\theta), \theta)$ and $(g(\theta), \theta)$. Write an expression for $w(\theta)$. Find $w_A$, the average value of $w(\theta)$ over the interval $0 \leq \theta \leq \frac{\pi}{2}$.
(d) Using the information from part (c), find the value of $\theta$ for which $w(\theta) = w_A$. Is the function $w(\theta)$ increasing or decreasing at that value of $\theta$? Give a reason for your answer.