12.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cb92f7b6-2ba5-4703-9595-9ba8570fc52b-21_645_935_301_589}
\captionsetup{labelformat=empty}
\caption{Figure 5}
\end{figure}
Figure 5 shows part of the curve \(C\) with parametric equations
$$x = 2 \cos \theta \quad y = \sin 2 \theta \quad 0 \leq \theta \leq \frac { \pi } { 2 }$$
The region \(R\), shown shaded in figure 5, is bounded by the curve \(C\), the line \(x = \sqrt { 2 }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid revolution.
a. Show that the volume of the solid of revolution formed is given by the integral.
$$k \int _ { \frac { \pi } { 4 } } ^ { \frac { \pi } { 2 } } \sin ^ { 3 } \theta \cos ^ { 2 } \theta \mathrm {~d} \theta$$
where \(k\) is a constant.
\includegraphics[max width=\textwidth, alt={}, center]{cb92f7b6-2ba5-4703-9595-9ba8570fc52b-22_164_1148_54_118}
b. Hence, find the exact value for this volume, giving your answer in the form \(p \pi \sqrt { 2 }\) where \(p\) is a constant.