7. The curve \(C\) has parametric equations
$$x = \cosh t + t , \quad y = \cosh t - t \quad 0 \leqslant t \leqslant \ln 3$$
- Show that
$$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = 2 \cosh ^ { 2 } t$$
The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is given by \(S\).
- Show that
$$S = 2 \pi \sqrt { 2 } \int _ { 0 } ^ { \ln 3 } \left( \cosh ^ { 2 } t - t \cosh t \right) d t$$
- Hence find the value of \(S\), giving your answer in the form
$$\frac { \pi \sqrt { 2 } } { 9 } ( a + b \ln 3 )$$
where \(a\) and \(b\) are constants to be determined.