Use integration by parts to find \(\int x \sin x \mathrm {~d} x\).
Using the substitution \(u = x ^ { 2 } + 5\), or otherwise, find \(\int x \sqrt { x ^ { 2 } + 5 } \mathrm {~d} x\).
The diagram shows the curve \(y = x ^ { 2 } - 9\) for \(x \geqslant 0\).
\includegraphics[max width=\textwidth, alt={}, center]{6890a681-2b7f-4853-a5f0-f88b7b435367-3_844_663_685_694}
The shaded region \(R\) is bounded by the curve, the lines \(y = 1\) and \(y = 2\), and the \(y\)-axis.
Find the exact value of the volume of the solid generated when the region \(R\) is rotated through \(360 ^ { \circ }\) about the \(\boldsymbol { y }\)-axis.