On , the cardioid lies outside the circle . Which of the following integrals…

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

Problem

0\le\theta\le\tfrac{\pi}{2}

, the cardioid

r=2+2\cos\theta

lies outside the circle

r=2

. Which of the following integrals represents the area of the region between the two curves on this interval? (a)

\displaystyle \int_{0}^{\pi/2}\bigl[(2+2\cos\theta)^2-2^2\bigr]\,d\theta

(b)

\displaystyle \tfrac12\int_{0}^{\pi/2}\bigl[(2+2\cos\theta)-2\bigr]^2\,d\theta

(c)

\displaystyle \tfrac12\int_{0}^{\pi/2}\bigl[(2+2\cos\theta)^2-2^2\bigr]\,d\theta

(d)

\displaystyle \tfrac12\int_{0}^{\pi}\bigl[(2+2\cos\theta)^2-2^2\bigr]\,d\theta