| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Standard +0.3 This is a straightforward application question requiring table completion by substituting into a given function, applying the trapezium rule formula (a standard C4 technique), simple area subtraction, and basic reasoning about concavity. All steps are routine with no novel problem-solving required, 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 Q6 [9]}}