Problem

Figure 1 shows a sketch of part of the curve with equation $y=e^{0.5x-2}$.

The region $R$, shown shaded in Figure 1, is bounded by the curve, the $x$-axis and the $y$-axis.

The region $R$ is rotated $360^\circ$ about the $x$-axis to form a solid of revolution.

Show that the volume of this solid can be written in the form $a\ln 2+b$, where $a$ and $b$ are constants to be found.