2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{cfc4afbd-3353-4f9f-b954-cb5178ebcf6c-06_624_872_210_543}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of the curve \(C\) with parametric equations
$$x = \ln ( \sec \theta + \tan \theta ) - \sin \theta \quad y = \cos \theta \quad 0 \leqslant \theta \leqslant \frac { \pi } { 4 }$$
The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis and is used to form a solid of revolution \(S\).
Using calculus, show that the total surface area of \(S\) is given by
$$\frac { \pi } { 2 } ( p + q \sqrt { 2 } )$$
where \(p\) and \(q\) are integers to be determined.