14.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{29b56d51-120a-4275-a761-8b8aed7bca54-48_506_812_219_571}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{figure}
Figure 6 shows a sketch of the curve \(C\) with parametric equations
$$x = 8 \cos ^ { 3 } \theta , \quad y = 6 \sin ^ { 2 } \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$
Given that the point \(P\) lies on \(C\) and has parameter \(\theta = \frac { \pi } { 3 }\)
- find the coordinates of \(P\).
The line \(l\) is the normal to \(C\) at \(P\).
- Show that an equation of \(l\) is \(y = x + 3.5\)
The finite region \(S\), shown shaded in Figure 6, is bounded by the curve \(C\), the line \(l\), the \(y\)-axis and the \(x\)-axis.
- Show that the area of \(S\) is given by
$$4 + 144 \int _ { 0 } ^ { \frac { \pi } { 3 } } \left( \sin \theta \cos ^ { 2 } \theta - \sin \theta \cos ^ { 4 } \theta \right) d \theta$$
- Hence, by integration, find the exact area of \(S\).
(Solutions based entirely on graphical or numerical methods are not acceptable.)