5.
\begin{figure}[h]
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{964070ca-a2c0-4935-8a5b-f1f656495f2e-3_771_1049_251_477}
\end{figure}
Figure 1 shows part of the curve with equation \(y = 1 + \frac { 1 } { 2 \sqrt { x } }\). The shaded region \(R\), bounded by the curve, that \(x\)-axis and the lines \(x = 1\) and \(x = 4\), is rotated through \(360 ^ { \circ }\) about the \(x\)-axis. Using integration, show that the volume of the solid generated is \(\pi \left( 5 + \frac { 1 } { 2 } \ln 2 \right)\).
(8)