9.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{216f5735-a7ad-4d70-9da9-ae1f098a97d9-20_714_714_269_616}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows a sketch of part of the curve with parametric equations
$$x = \tan \theta \quad y = 2 \sin 2 \theta \quad \theta \geqslant 0$$
The finite region, shown shaded in Figure 3, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \sqrt { 3 }\)
The region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
- Show that the exact volume of this solid of revolution is given by
$$\int _ { 0 } ^ { k } p ( 1 - \cos 2 \theta ) d \theta$$
where \(p\) and \(k\) are constants to be found.
- Hence find, by algebraic integration, the exact volume of this solid of revolution.