6.
\begin{figure}[h]
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\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{07bc7f2d-c2b9-4502-91cd-a76afb1ca6c0-5_809_1226_201_303}
\end{figure}
Figure 2 shows the cross-section of a road tunnel and its concrete surround. The curved section of the tunnel is modelled by the curve with equation \(y = 8 \sqrt { \left( \sin \frac { \pi x } { 10 } \right) }\), in the interval \(0 \leq x \leq\) 10. The concrete surround is represented by the shaded area bounded by the curve, the \(x\)-axis and the lines \(x = - 2 , x = 12\) and \(y = 10\). The units on both axes are metres.
- Using this model, copy and complete the table below, giving the values of \(y\) to 2 decimal places.
The area of the cross-section of the tunnel is given by \(\int _ { 0 } ^ { 10 } y \mathrm {~d} x\).
- Estimate this area, using the trapezium rule with all the values from your table.
- Deduce an estimate of the cross-sectional area of the concrete surround.
- State, with a reason, whether your answer in part (c) over-estimates or under-estimates the true value.
(2)