7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fe07afad-9cfc-48c0-84f1-5717f81977d4-20_473_313_244_350}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{fe07afad-9cfc-48c0-84f1-5717f81977d4-20_390_627_246_970}
\captionsetup{labelformat=empty}
\caption{Figure 4}
\end{figure}
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
$$\mathrm { f } ( x ) = \frac { 1 } { 4 } ( 4 - x ) \mathrm { e } ^ { x } \quad 0 \leqslant x \leqslant 4$$
- Show that the volume, \(V \mathrm {~cm} ^ { 3 }\), of the doorknob is given by
$$V = K \int _ { 0 } ^ { 4 } \left( x ^ { 2 } - 8 x + 16 \right) \mathrm { e } ^ { 2 x } \mathrm {~d} x$$
where \(K\) is a constant to be found.
- Hence, find the exact value of the volume of the doorknob.
Give your answer in the form \(p \pi \left( \mathrm { e } ^ { q } + r \right) \mathrm { cm } ^ { 3 }\) where \(p , q\) and \(r\) are simplified rational numbers to be found.