By using integration by parts, find \(\int x \mathrm { e } ^ { 6 x } \mathrm {~d} x\).
(4 marks)
The diagram shows part of the curve with equation \(y = \sqrt { x } \mathrm { e } ^ { 3 x }\).
\includegraphics[max width=\textwidth, alt={}, center]{d3c66c34-b09c-4223-8383-cf0a68419bf9-4_547_846_536_591}
The shaded region \(R\) is bounded by the curve \(y = \sqrt { x } \mathrm { e } ^ { 3 x }\), the line \(x = 1\) and the \(x\)-axis from \(x = 0\) to \(x = 1\).
Find the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis, giving your answer in the form \(\pi \left( p \mathrm { e } ^ { 6 } + q \right)\), where \(p\) and \(q\) are rational numbers.
(3 marks)