8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cd958ff3-ed4e-4bd7-aa4b-339da6d618a6-28_721_714_255_616}
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\caption{Figure 4}
\end{figure}
Diagram not drawn to scale
Figure 4 shows a sketch of part of the curve \(C\) with parametric equations
$$x = 3 \theta \sin \theta , \quad y = \sec ^ { 3 } \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$
The point \(P ( k , 8 )\) lies on \(C\), where \(k\) is a constant.
- Find the exact value of \(k\).
The finite region \(R\), shown shaded in Figure 4, is bounded by the curve \(C\), the \(y\)-axis, the \(x\)-axis and the line with equation \(x = k\).
- Show that the area of \(R\) can be expressed in the form
$$\lambda \int _ { \alpha } ^ { \beta } \left( \theta \sec ^ { 2 } \theta + \tan \theta \sec ^ { 2 } \theta \right) \mathrm { d } \theta$$
where \(\lambda , \alpha\) and \(\beta\) are constants to be determined.
- Hence use integration to find the exact value of the area of \(R\).