7. The curve \(C\) has parametric equations
$$x = 3 t ^ { 4 } , \quad y = 4 t ^ { 3 } , \quad 0 \leqslant t \leqslant 1$$
The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis. The area of the curved surface generated is \(S\).
- Show that
$$S = k \pi \int _ { 0 } ^ { 1 } t ^ { 5 } \left( t ^ { 2 } + 1 \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} t$$
where \(k\) is a constant to be found.
- Use the substitution \(u ^ { 2 } = t ^ { 2 } + 1\) to find the value of \(S\), giving your answer in the form \(p \pi ( 11 \sqrt { 2 } - 4 )\) where \(p\) is a rational number to be found.