7. The curve \(C\) has parametric equations
$$x = 3 t ^ { 2 } , \quad y = 12 t , \quad 0 \leqslant t \leqslant 4$$
The curve \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
- Show that the area of the surface generated is
$$\pi ( a \sqrt { 5 } + b )$$
where \(a\) and \(b\) are constants to be found.
- Show that the length of the curve \(C\) is given by
$$k \int _ { 0 } ^ { 4 } \sqrt { \left( t ^ { 2 } + 4 \right) } \mathrm { d } t$$
where \(k\) is a constant to be found.
- Use the substitution \(t = 2 \sinh \theta\) to show that the exact value of the length of the curve \(C\) is
$$24 \sqrt { 5 } + 12 \ln ( 2 + \sqrt { 5 } )$$