9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e583bf92-d6a9-4f1a-b3c8-372afa6e0a0e-26_543_604_255_733}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
The curve \(C\), shown in Figure 3, has equation
$$y = \frac { x ^ { - \frac { 1 } { 4 } } } { \sqrt { 1 + x } ( \arctan \sqrt { x } ) }$$
The region \(R\), shown shaded in Figure 3, is bounded by \(C\), the line with equation \(x = 3\), the \(x\)-axis and the line with equation \(x = \frac { 1 } { 3 }\)
The region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis to form a solid.
Using the substitution \(\tan u = \sqrt { x }\)
- show that the volume \(V\) of the solid formed is given by
$$k \int _ { a } ^ { b } \frac { 1 } { u ^ { 2 } } \mathrm {~d} u$$
where \(k , a\) and \(b\) are constants to be found.
- Hence, using algebraic integration, find the value of \(V\) in simplest form.