The curve \(C\) has parametric equations
\(x=t^{2}, \quad y=t-\frac{1}{3} t^{3}, \quad \text { for } 0 \leqslant t \leqslant 1\)

Find the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

Find the coordinates of the centroid of the region bounded by \(C\), the \(x\)-axis and the line \(x=1\).