Mathematics

Problem

What is the exact expression for the area of the region that lies between the two polar curves

r=f(\theta)

(outer) and

r=g(\theta)

(inner) for

\theta

[a,b]

, where

f(\theta)\ge g(\theta)\ge 0

on that interval? a)

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

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

\displaystyle \int_{a}^{b} \bigl[f(\theta)-g(\theta)\bigr] \,d\theta

\displaystyle \tfrac12\bigl[f(\theta)^2-g(\theta)^2\bigr]\Bigl|_{a}^{b}