2 A curve has parametric equations
$$x = t - \frac { 1 } { 3 } t ^ { 3 } , \quad y = t ^ { 2 }$$
- Show that
$$\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + t ^ { 2 } \right) ^ { 2 }$$
- The arc of the curve between \(t = 1\) and \(t = 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
Show that \(S\), the surface area generated, is given by \(S = k \pi\), where \(k\) is a rational number to be found.