The curves and intersect at and

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

Problem

The curves

r=1+2\sin\theta

and

r=2

intersect at

\theta=\pi/6

and

\theta=5\pi/6

. What is the exact area of the region they enclose for

0\le\theta\le 2\pi

? (a)

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

(b)

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

(c)

\dfrac{\pi}{3}+5\sqrt3

(d)

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