7.
\section*{Figure 1}
\includegraphics[max width=\textwidth, alt={}]{a02ed733-d5ee-45a7-a39a-e40f3a36e659-16_634_1025_191_479}
Figure 1 shows the finite region \(R\), which is bounded by the curve \(y = x \mathrm { e } ^ { x }\), the line \(x = 1\), the line \(x = 3\) and the \(x\)-axis.
The region \(R\) is rotated through 360 degrees about the \(x\)-axis.
Use integration by parts to find an exact value for the volume of the solid generated.
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