The curve \(C\) is defined parametrically by
$$x = 18 t - t ^ { 2 } \quad \text { and } \quad y = 8 t ^ { \frac { 3 } { 2 } }$$
where \(0 < t \leqslant 4\).
- Show that at all points of \(C\),
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 3 ( 9 + t ) } { 2 t ^ { \frac { 1 } { 2 } } ( 9 - t ) ^ { 3 } }$$
- Show that the mean value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) with respect to \(x\) over the interval \(0 < x \leqslant 56\) is \(\frac { 3 } { 70 }\).
- Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, showing full working.