5 Fig. 2 shows the curve \(y = \sqrt { } 1 + \mathrm { e } ^ { 2 x }\).
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{8d786d33-c5c2-44a6-8273-7a3e43e552ef-4_434_873_306_665}
\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 \(\frac { 1 } { 2 } \pi \left( 1 + \mathrm { e } ^ { 2 } \right)\).