2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{38487750-8c0f-4c3d-a019-5213ed2866eb-04_616_764_246_584}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{figure}
Figure 1 shows a sketch of part of the curve with equation
$$y = 5 \cosh x - 6 \sinh x$$
The curve crosses the \(x\)-axis at the point \(A\).
- Find the exact value of the \(x\) coordinate of the point \(A\), giving your answer as a natural logarithm.
- Show that
$$( 5 \cosh x - 6 \sinh x ) ^ { 2 } \equiv a \cosh 2 x + b \sinh 2 x + c$$
where \(a , b\) and \(c\) are constants to be found.
The finite region \(R\), bounded by the curve and the coordinate axes, is shown shaded in Figure 1.
The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
- Use calculus to find the volume of the solid generated, giving your answer as an exact multiple of \(\pi\).