4 A curve \(C\) is given parametrically by the equations
$$x = \frac { 1 } { 2 } \cosh 2 t , \quad y = 2 \sinh t$$
- Express
$$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 }$$
in terms of \(\cosh t\).
- The arc of \(C\) from \(t = 0\) to \(t = 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
- Show that \(S\), the area of the curved surface generated, is given by
$$S = 8 \pi \int _ { 0 } ^ { 1 } \sinh t \cosh ^ { 2 } t \mathrm {~d} t$$
- Find the exact value of \(S\).