The curves and intersect at two angles in

MATHEMATICS · Single correct · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

The curves

r_1(\theta)=4\cos\theta

and

r_2(\theta)=2\cos\theta+1

intersect at two angles in

[0,2\pi)

. The bounded region lying between the curves and contained in the interval

0\le \theta\le\tfrac{\pi}{3}

is shaded. What is the exact area of this shaded region? (a)

\dfrac{5\pi}{3}-\dfrac{\sqrt3}{2}

(b)

\dfrac{5\pi}{12}-\dfrac{\sqrt3}{8}

(c)

\dfrac{5\pi}{6}-\dfrac{\sqrt3}{4}

(d)

\dfrac{5\pi}{6}+\dfrac{\sqrt3}{2}