3 At any point \(( x , y )\) on the curve \(C\), $$\frac { \mathrm { d } x } { \mathrm {~d} t } = t \sqrt { } \left( t ^ { 2 } + 4 \right) \quad \text { and } \quad \frac { \mathrm { d } y } { \mathrm {~d} t } = - t \sqrt { } \left( 4 - t ^ { 2 } \right)$$ where the parameter \(t\) is such that \(0 \leqslant t \leqslant 2\). Show that the length of \(C\) is \(4 \sqrt { } 2\). Given that \(y = 0\) when \(t = 2\), determine the area of the surface generated when \(C\) is rotated through one complete revolution about the \(x\)-axis, leaving your answer in an exact form.