5 A curve has parametric equations \(x = \sec \theta , y = 2 \tan \theta\).
- Given that the derivative of \(\sec \theta\) is \(\sec \theta \tan \theta\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \operatorname { cosec } \theta\).
- Verify that the cartesian equation of the curve is \(y ^ { 2 } = 4 x ^ { 2 } - 4\).
Fig. 5 shows the region enclosed by the curve and the line \(x = 2\). This region is rotated through \(180 ^ { \circ }\) about the \(x\)-axis.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{132ae754-bd4c-4819-80ef-4823ac2ead4f-02_545_853_1738_607}
\captionsetup{labelformat=empty}
\caption{Fig. 5}
\end{figure} - Find the volume of revolution produced, giving your answer in exact form.