The curve $C$ has equation $y = 2 \sec x$, for $0 \leqslant x \leqslant \frac { 1 } { 4 } \pi$. Show that the arc length $s$ of $C$ is given by

$$S = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 2 } x - 1 \right) d x$$

Find the exact value of $s$.

The surface area generated when $C$ is rotated through $2 \pi$ radians about the $x$-axis is denoted by $S$. Show that\\
(i) $S = 4 \pi \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \left( 2 \sec ^ { 3 } x - \sec x \right) \mathrm { d } x$,\\
(ii) $\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x \tan x ) = 2 \sec ^ { 3 } x - \sec x$.

Hence find the exact value of $S$.

\hfill \mbox{\textit{CAIE FP1 2013 Q11 EITHER}}