11.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{a377da06-a968-438c-bec2-ae55283dae47-36_601_1140_242_402}
\captionsetup{labelformat=empty}
\caption{Figure 3}
\end{figure}
- By writing \(\sec \theta\) as \(\frac { 1 } { \cos \theta }\), show that when \(x = 3 \sec \theta\),
$$\frac { \mathrm { d } x } { \mathrm {~d} \theta } = 3 \sec \theta \tan \theta$$
Figure 3 shows a sketch of part of the curve \(C\) with equation
$$y = \frac { \sqrt { x ^ { 2 } - 9 } } { x } \quad x \geqslant 3$$
The finite region \(R\), shown shaded in Figure 3, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 6\)
- Use the substitution \(x = 3 \sec \theta\) to find the exact value of the area of \(R\). [Solutions based entirely on graphical or numerical methods are not acceptable.]