8.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c20a4592-74c6-4f58-b63b-984b171b1bfd-28_552_380_264_468}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c20a4592-74c6-4f58-b63b-984b171b1bfd-28_524_446_274_1151}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{figure}
Figure 1 shows a French horn with a detachable bell section.
The shape of the bell section can be modelled by rotating an exponential curve through \(360 ^ { \circ }\) about the \(x\)-axis, where units are centimetres.
The model uses the curve shown in Figure 2, with equation
$$y = \frac { 9 } { 2 } e ^ { \frac { 1 } { 9 } x } \quad 0 \leqslant x \leqslant 9$$
- Show that, according to this model, the external surface area of the bell section is given by
$$K \int _ { 0 } ^ { 9 } \mathrm { e } ^ { \frac { 1 } { 9 } x } \sqrt { 4 + \mathrm { e } ^ { \frac { 2 } { 9 } x } } \mathrm {~d} x$$
where \(K\) is a real constant to be determined.
- Use the substitution \(u = e ^ { \frac { 1 } { 9 } x }\) to show that
$$\int _ { 0 } ^ { 9 } \mathrm { e } ^ { \frac { 1 } { 9 } x } \sqrt { 4 + \mathrm { e } ^ { \frac { 2 } { 9 } x } } \mathrm {~d} x = 9 \int _ { a } ^ { b } \frac { 2 u + u ^ { 3 } } { \sqrt { 4 u ^ { 2 } + u ^ { 4 } } } \mathrm {~d} u + 18 \int _ { a } ^ { b } \frac { 1 } { \sqrt { 4 + u ^ { 2 } } } \mathrm {~d} u$$
where \(a\) and \(b\) are constants to be determined.
Hence, using algebraic integration,
- determine, according to the model, the external surface area of the bell section of the horn, giving your answer to 3 significant figures.