3 A curve has parametric equations $$x = a \left( 1 - \cos ^ { 3 } \theta \right) , \quad y = a \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { \pi } { 3 }$$ where \(a\) is a positive constant.
The arc length from the origin to a general point on the curve is denoted by \(s\), and \(\psi\) is the acute angle defined by \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }\).

1. Express \(s\) and \(\psi\) in terms of \(\theta\), and hence show that the intrinsic equation of the curve is $$s = \frac { 3 } { 2 } a \sin ^ { 2 } \psi$$
2. For the point on the curve given by \(\theta = \frac { \pi } { 6 }\), find the radius of curvature and the coordinates of the centre of curvature.
3. Find the area of the curved surface generated when the curve is rotated through \(2 \pi\) radians about the \(y\)-axis.