In polar coordinates

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

Problem

In polar coordinates, let two curves be defined by

r=f(\theta)

and

r=g(\theta)

for

\alpha\le\theta\le\beta

, with

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

on that interval. Which of the following expressions gives the area of the region between the two curves? (a)

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

(b)

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

(c)

\displaystyle \tfrac12\int_{\alpha}^{\beta}\bigl[f(\theta)^2+g(\theta)^2\bigr]\,d\theta

(d)

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