6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0f2d48ab-1f61-4fb9-b35a-25d684dbd50f-10_454_602_255_479}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows the curve with parametric equations
$$x = 3 \sin t , \quad y = 2 \sin 2 t , \quad 0 \leq t < \pi .$$
The curve meets the \(x\)-axis at the origin, \(O\), and at the point \(A\).
- Find the value of \(t\) at \(O\) and the value of \(t\) at \(A\).
The region enclosed by the curve is rotated through \(\pi\) radians about the \(x\)-axis.
- Show that the volume of the solid formed is given by
$$\int _ { 0 } ^ { \frac { \pi } { 2 } } 12 \pi \sin ^ { 2 } 2 t \cos t \mathrm {~d} t$$
- Using the substitution \(u = \sin t\), or otherwise, evaluate this integral, giving your answer as an exact multiple of \(\pi\).
6. continued