The curve \(C\) has equation \(y = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\) for \(0 \leqslant x \leqslant 4\).
- The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x = 4\). Find, in terms of e, the coordinates of the centroid of the region \(R\).
- Show that \(\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \left( \mathrm { e } ^ { x } + \mathrm { e } ^ { - x } \right)\), where \(s\) denotes the arc length of \(C\), and find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.