Pappus's Theorem Let \(A\) be the area of the region \(\mathcal{D}\) between two graphs \(y=g_{1}(x)\) and \(y=g_{2}(x)\) over the interval \([a, b]\), where \(g_{2}(x) \geq g_{1}(x) \geq 0\). Prove Pappus's Theorem: The volume of the solid obtained by revolving \(\mathcal{D}\) about the \(x\)-axis is \(V=2 \pi A \bar{y}\), where \(\bar{y}\) is the \(y\)-coordinate of the centroid of \(\mathcal{D}\) (the average of the \(y\)-coordinate). Hint: Show that

\[A \bar{y}=\int_{x=a}^{b} \int_{y=g_{1}(x)}^{g_{2}(x)} y d y d x\]