8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fb1924cc-9fa3-4fde-ba4d-6fb095f7f70b-11_639_972_228_484}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows the curve \(C\) with parametric equations
$$x = 8 \cos t , \quad y = 4 \sin 2 t , \quad 0 \leqslant t \leqslant \frac { \pi } { 2 } .$$
The point \(P\) lies on \(C\) and has coordinates \(( 4,2 \sqrt { } 3 )\).
- Find the value of \(t\) at the point \(P\).
The line \(l\) is a normal to \(C\) at \(P\).
- Show that an equation for \(l\) is \(y = - x \sqrt { 3 } + 6 \sqrt { 3 }\).
The finite region \(R\) is enclosed by the curve \(C\), the \(x\)-axis and the line \(x = 4\), as shown shaded in Figure 3.
- Show that the area of \(R\) is given by the integral \(\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } 64 \sin ^ { 2 } t \cos t \mathrm {~d} t\).
- Use this integral to find the area of \(R\), giving your answer in the form \(a + b \sqrt { } 3\), where \(a\) and \(b\) are constants to be determined.