Problem

Figure 1 shows a sketch of the curve $C$ with parametric equations

$$x=\ln(\sec\theta+\tan\theta)-\sin\theta \qquad y=\cos\theta \qquad 0\le\theta\le\frac{\pi}{4}$$

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

$$\frac{\pi}{2}(p+q\sqrt2)$$

where $p$ and $q$ are integers to be determined.