Problem

In this question you must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

(a) Find

$$\int x^2\ln x\,dx$$

Figure 3 shows a sketch of part of the curve with equation

$$y=x\ln x \qquad x>0$$

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 $x=e$.

This region is rotated through $2\pi$ radians about the $x$-axis to form a solid of revolution.

(b) Find the exact volume of the solid formed, giving your answer in simplest form.