4.
\includegraphics[max width=\textwidth, alt={}, center]{d6befd60-de40-41b6-8ae5-48656dbca40c-3_535_1027_276_577}
The diagram above shows a sketch of the curve \(C\) with polar equation
$$r = 4 \sin \theta \cos ^ { 2 } \theta , \quad 0 \leq \theta < \frac { \pi } { 2 }$$
The tangent to \(C\) at the point \(P\) is perpendicular to the initial line.
- Show that \(P\) has polar coordinates \(\left( \frac { 3 } { 2 } , \frac { \pi } { 6 } \right)\).
The point \(Q\) on \(C\) has polar coordinates \(\left( \sqrt { 2 } , \frac { \pi } { 4 } \right)\).
The shaded region \(R\) is bounded by \(O P , O Q\) and \(C\), as shown in the diagram above. - Show that the area of \(R\) is given by
$$\int _ { \frac { \pi } { 6 } } ^ { \frac { \pi } { 4 } } \left( \sin ^ { 2 } 2 \theta \cos 2 \theta + \frac { 1 } { 2 } - \frac { 1 } { 2 } \cos 4 \theta \right) \mathrm { d } \theta$$
- Hence, or otherwise, find the area of \(R\), giving your answer in the form \(a + b \pi\), where \(a\) and \(b\) are rational numbers.
(Total 14 marks)