For the region bounded by the outer curve and the inner curve on the interval with

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

Problem

For the region bounded by the outer curve

r=f(\theta)

and the inner curve

r=g(\theta)

on the interval

\alpha\le \theta\le\beta

with

f(\theta)\ge g(\theta)\ge0

, which integral gives the exact area of the region? (a)

\displaystyle \int_{\alpha}^{\beta}\!\bigl(f(\theta)^2-g(\theta)^2\bigr)\,d\theta

(b)

\displaystyle \tfrac12\int_{\alpha}^{\beta}\!\bigl(f(\theta)-g(\theta)\bigr)^2\,d\theta

(c)

\displaystyle \tfrac12\int_{\alpha}^{\beta}\!\bigl(f(\theta)^2-g(\theta)^2\bigr)\,d\theta

(d)

\displaystyle \tfrac12\int_{\alpha}^{\beta}\!f(\theta)^2\,d\theta