Let \(F(x,y) = 8-y\) and \(R\) be the region enclosed by the circles \(r=\cos (\theta )\) and \(r=3\cos (\theta )\). Use polar coordinates to set-up and evaluate \(\iint _R F(x,y) \d A\).
\begin{align*} \iint _R F(x,y) \d A &= \int _{\answer {0}}^{\answer {\pi }}\int _{\answer {\cos (\theta )}}^{\answer {3\cos (\theta )}} \answer {(8-r\sin (\theta ))r} \d r \d \theta \\ &= \answer {16\pi } \end{align*}