Find the volume $V$ of the solid of revolution generated by revolving the region bounded by the graph of $y=2x-2x^{2}$ and the $x$-axis about the line $x=2$.

Solution The region bounded by the graph of $y=2x-2x^{2}$ and the $x$-axis is illustrated in Figure 37(a). A typical shell formed by revolving the region about the line $x=2$, as shown in Figure 37(b), has an average radius of $2-u_{i}$, height $f(u_{i}) = 2u_{i}-2u_{i}^{2}$, and thickness $\Delta x$. The solid of revolution is depicted in Figure 37(c).

Figure 37

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The volume $V$ of the solid is $$\begin{eqnarray*} V&=&2\pi \displaystyle \int_{0}^{1}(2-x)(2x-2x^{2})~{\it dx}=4\pi \displaystyle \int_{0}^{1}(x^{3}-3x^{2}+2x)~{\it dx}\\[4pt] &=&4\pi \left[ \dfrac{x^{4}}{4}-x^{3}+x^{2} \right] _{0}^{1}=\pi \hbox{ cubic units} \end{eqnarray*}$$