2.
\includegraphics[max width=\textwidth, alt={}]{49d985bf-7c94-4a54-88c1-c0084cd94000-1_563_833_532_513}
The diagram shows the curve with equation \(y = \frac { 3 x + 1 } { \sqrt { x } } , x > 0\).
The shaded region is bounded by the curve, the \(x\)-axis and the lines \(x = 1\) and \(x = 3\).
Find the volume of the solid formed when the shaded region is rotated through four right angles about the \(x\)-axis, giving your answer in the form \(\pi ( a + \ln b )\), where \(a\) and \(b\) are integers.