| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.3 This is a straightforward application of the trapezium rule with clearly defined steps: substitute values into a given function, apply the standard trapezium rule formula, then perform simple arithmetic to find the concrete area. While it requires careful calculation across multiple parts, it involves only routine techniques with no conceptual challenges or novel problem-solving, making it slightly easier than average. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 2 | 4 | 6 | 8 | 10 |
| \(y\) | 0 | 6.13 | 0 |
\includegraphics{figure_2}
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 $y = 8\sqrt{\sin \frac{\pi x}{10}}$, 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.
\begin{enumerate}[label=(\alph*)]
\item Using this model, copy and complete the table below, giving the values of $y$ to 2 decimal places.
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
$x$ & 0 & 2 & 4 & 6 & 8 & 10 \\
\hline
$y$ & 0 & 6.13 & & & & 0 \\
\hline
\end{tabular}
[2]
\end{enumerate}
The area of the cross-section of the tunnel is given by $\int_0^{10} y \, dx$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Estimate this area, using the trapezium rule with all the values from your table. [4]
\item Deduce an estimate of the cross-sectional area of the concrete surround. [1]
\item State, with a reason, whether your answer in part $(c)$ over-estimates or under-estimates the true value. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q27 [9]}}