10.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-34_599_923_322_571}
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\caption{Figure 3}
\end{figure}
Figure 3 shows part of the curve \(C\) with parametric equations
$$x = \tan \theta , \quad y = \sin \theta , \quad 0 \leqslant \theta \leqslant \frac { \pi } { 2 }$$
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.
\includegraphics[max width=\textwidth, alt={}, center]{e375f6ad-4a76-42a0-b7bf-ae47e5cbdaeb-39_61_29_2608_1886}