Find the length of the arc of the polar curve \(r = a ( 1 + \cos \theta )\) for which \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
A curve \(C\) has cartesian equation \(y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x }\).
The arc of \(C\) for which \(1 \leqslant x \leqslant 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a surface of revolution. Find the area of this surface.
For the point on \(C\) at which \(x = 2\),
show that the radius of curvature is \(\frac { 289 } { 64 }\),