78ed8c…: “Onto the lime line”

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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 $\iint _R F(x,y) \d A$ . $\begin{align*} \iint_R F(x,y) \d A &= \int_{\answer{0}}^{\answer{\pi}}\int_{\answer{\cos(\theta)}}^{\answer{3\cos(\theta)}} \answer{(8-r\sin(\theta))r} \d r \d \theta\\ &= \answer{16\pi} \end{align*}$