7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{9d513d77-b8f9-4223-832f-f566c5f50457-10_643_999_276_475}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
Figure 3 shows part of the curve \(C\) with parametric equations
$$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta < \frac { \pi } { 2 }$$
The point \(P\) lies on \(C\) and has coordinates \(\left( \sqrt { } 3 , \frac { 1 } { 2 } \sqrt { } 3 \right)\).
- Find the value of \(\theta\) at the point \(P\).
The line \(l\) is a normal to \(C\) at \(P\). The normal cuts the \(x\)-axis at the point \(Q\).
- Show that \(Q\) has coordinates \(( k \sqrt { } 3,0 )\), giving the value of the constant \(k\).
The finite shaded region \(S\) shown in Figure 3 is bounded by the curve \(C\), the line \(x = \sqrt { } 3\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
- Find the volume of the solid of revolution, giving your answer in the form \(p \pi \sqrt { } 3 + q \pi ^ { 2 }\), where \(p\) and \(q\) are constants.