3 The curve \(C\) has parametric equations \(x = 2 t ^ { 3 } - 6 t , y = 6 t ^ { 2 }\).
- Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 1\).
- Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
- Show that the equation of the normal to \(C\) at the point with parameter \(t\) is
$$y = \frac { 1 } { 2 } \left( \frac { 1 } { t } - t \right) x + 2 t ^ { 2 } + t ^ { 4 } + 3$$
- Find the cartesian equation of the envelope of the normals to \(C\).
- The point \(\mathrm { P } ( 64 , a )\) is the centre of curvature corresponding to a point on \(C\). Find \(a\).