Problem

Figure 3 shows the design of a doorknob.

The shape of the doorknob is formed by rotating the curve shown in Figure 4 through $360^\circ$ about the $x$-axis, where the units are centimetres.

The equation of the curve is given by

$$\mathrm{f}(x)=\frac{1}{4}(4-x)e^x \qquad 0\leq x\leq 4$$

(a) Show that the volume, $V$ cm$^3$, of the doorknob is given by

$$V=K\int_0^4 (x^2-8x+16)e^{2x}\,dx$$

where $K$ is a constant to be found.

(b) Hence, find the exact value of the volume of the doorknob.

Give your answer in the form $p\pi(e^q+r)$ cm$^3$ where $p$, $q$ and $r$ are simplified rational numbers to be found.