3 A curve \(C\) is defined parametrically by
$$x = \frac { t ^ { 2 } + 1 } { t } , \quad y = 2 \ln t$$
- Show that \(\left( \frac { \mathrm { d } x } { \mathrm {~d} t } \right) ^ { 2 } + \left( \frac { \mathrm { d } y } { \mathrm {~d} t } \right) ^ { 2 } = \left( 1 + \frac { 1 } { t ^ { 2 } } \right) ^ { 2 }\).
- The arc of \(C\) from \(t = 1\) to \(t = 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find the area of the surface generated, giving your answer in the form \(\pi ( m \ln 2 + n )\), where \(m\) and \(n\) are integers.
[0pt]
[5 marks]