Using the Disk Method: Revolving About the \(x\)-Axis

Find the volume of the solid of revolution generated by revolving the region bounded by the graph of \(y=x^{3}\), the \(x\)-axis, and the lines \(x=-1\) and \(x=2\) about the \(x\)-axis.

Solution Figure 18(a) shows the graph of the region to be revolved about the \(x\)-axis. Figure 18(b) illustrates a typical disk and Figure 18(c) shows the solid of revolution.

Using the disk method, the volume \(V\) of the solid of revolution is \begin{equation*} V=\pi \int_{-1}^{2}x^{6}{\it dx}=\pi \left[ \frac{x^{7}}{7}\right] _{-1}^{2}=\frac{ \pi }{7}(128+1)=\frac{129}{7}\pi \hbox{ cubic units} \end{equation*}