The curve \(C\) has equation \(y = \frac { 1 } { 3 } x ^ { \frac { 1 } { 2 } } ( 3 - x )\), for \(0 \leqslant x \leqslant 3\). Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 3\). Show that $$\frac { \mathrm { d } s } { \mathrm {~d} x } = \frac { 1 } { 2 } \left( x ^ { - \frac { 1 } { 2 } } + x ^ { \frac { 1 } { 2 } } \right)$$ where \(s\) denotes arc length, and find the arc length of \(C\). Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.