A curve has equation \(y = \frac { 2 } { 3 } x ^ { \frac { 3 } { 2 } }\), for \(x \geqslant 0\). The arc of the curve joining the origin to the point where \(x = 3\) is denoted by \(R\).
- Find the length of \(R\).
- Find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis, the line \(x = 3\) and \(R\).
- Show that the area of the surface generated when \(R\) is rotated through one revolution about the \(y\)-axis is \(\frac { 232 } { 15 } \pi\).