The circle and the limaçon intersect at

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

Problem

The circle

r=2

and the limaçon

r=4\cos\theta

intersect at

\theta=\pm\tfrac{\pi}{3}

. Determine the exact area of the region that lies inside

r=4\cos\theta

and outside

r=2

. (a)

\displaystyle \frac{8\pi}{3}+4\sqrt{3}

(b)

\displaystyle \frac{2\pi}{3}+\sqrt{3}

(c)

\displaystyle \frac{4\pi}{3}+2\sqrt{3}

(d)

\displaystyle \frac{4\pi}{3}-2\sqrt{3}