16.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{129adfbb-98fa-4e88-b636-7b4d111f3349-46_770_999_242_534}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{figure}
Figure 6 shows a sketch of the curve \(C\) with parametric equations
$$x = 8 \sin ^ { 2 } t \quad y = 2 \sin 2 t + 3 \sin t \quad 0 \leqslant t \leqslant \frac { \pi } { 2 }$$
The region \(R\), shown shaded in Figure 6, is bounded by \(C\), the \(x\)-axis and the line with equation \(x = 4\)
- Show that the area of \(R\) is given by
$$\int _ { 0 } ^ { a } \left( 8 - 8 \cos 4 t + 48 \sin ^ { 2 } t \cos t \right) \mathrm { d } t$$
where \(a\) is a constant to be found.
- Hence, using algebraic integration, find the exact area of \(R\).