Use the substitution \(x = \cosh ^ { 2 } u\) to find \(\int \sqrt { \frac { x } { x - 1 } } \mathrm {~d} x\), giving your answer in the form \(\mathrm { f } ( x ) + \ln ( \mathrm { g } ( x ) )\).
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Hence calculate the exact area of the region between the curve \(y = \sqrt { \frac { x } { x - 1 } }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) (see diagram).
What can you say about the volume of the solid of revolution obtained when the region defined in part (ii) is rotated completely about the \(x\)-axis? Justify your answer.