The curve \(C\) has parametric equations
\(x=2 \cosh t, \quad y=\frac{3}{2} t-\frac{1}{4} \sinh 2 t, \text { for } 0 \leqslant t \leqslant 1 .\)
(a) Find \(\frac{\mathrm{d} x}{\mathrm{~d} t}\) and show that \(\frac{\mathrm{d} y}{\mathrm{~d} t}=1-\sinh ^{2} t\).

The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(A\).
(b) (i) Show that \(A=\pi \int_{0}^{1}\left(\frac{3}{2} t-\frac{1}{4} \sinh 2 t\right)(1+\cosh 2 t) \mathrm{d} t\).
(ii) Hence find \(A\) in terms of \(\pi, \sinh 2\) and \(\cosh 2\).