Edexcel FP3 2013 June — Question 7

Exam BoardEdexcel
ModuleFP3 (Further Pure Mathematics 3)
Year2013
SessionJune
TopicHyperbolic functions

7. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{094b3c91-1460-44a2-b9d6-4de90d3adfa0-13_593_1292_118_328} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} The curves shown in Figure 1 have equations $$y = 6 \cosh x \text { and } y = 9 - 2 \sinh x$$
  1. Using the definitions of \(\sinh x\) and \(\cosh x\) in terms of \(\mathrm { e } ^ { x }\), find exact values for the \(x\)-coordinates of the two points where the curves intersect. The finite region between the two curves is shown shaded in Figure 1.
  2. Using calculus, find the area of the shaded region, giving your answer in the form \(a \ln b + c\), where \(a , b\) and \(c\) are integers.