Find \(\int x ^ { 2 } \ln x \mathrm {~d} x\)
Figure 3 shows a sketch of part of the curve with equation
$$y = x \ln x \quad x > 0$$
The region \(R\), shown shaded in Figure 3, lies entirely above the \(x\)-axis and is bounded by the curve, the \(x\)-axis and the line with equation \(x = \mathrm { e }\).
This region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
Find the exact volume of the solid formed, giving your answer in simplest form.
\section*{8. In this question you must show all stages of your working.
In this question you must show all stages of your working.}