2 Fig. 2 shows the curve \(y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8ad99e2a-4cef-40b3-af8d-673b97536227-02_432_873_587_635}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{figure}
The region bounded by the curve, the \(x\)-axis, the \(y\)-axis and the line \(x = 1\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
Show that the volume of the solid of revolution produced is \(\frac { 1 } { 2 } \pi \left( 1 + \mathrm { e } ^ { 2 } \right)\).