| Exam Board | OCR MEI |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2016 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Trapezium rule for applications |
| Difficulty | Standard +0.3 This is a straightforward applied calculus question requiring standard techniques: trapezium rule (routine procedure), checking if a point lies below a curve (simple substitution), and polynomial integration. All steps are mechanical with no novel problem-solving required, making it slightly easier than average for A-level. |
| Spec | 1.08b Integrate x^n: where n != -1 and sums1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 |
| \(y\) | 0 | 4.0 | 4.9 | 5.0 | 4.9 | 4.0 | 0 |
Fig. 9 shows the cross-section of a straight, horizontal tunnel. The $x$-axis from 0 to 6 represents the floor of the tunnel.
\includegraphics{figure_9}
With axes as shown, and units in metres, the roof of the tunnel passes through the points shown in the table.
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$x$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline
$y$ & 0 & 4.0 & 4.9 & 5.0 & 4.9 & 4.0 & 0 \\
\hline
\end{tabular}
The length of the tunnel is 50 m.
\begin{enumerate}[label=(\roman*)]
\item Use the trapezium rule with 6 strips to estimate the area of cross-section of the tunnel. Hence estimate the volume of earth removed in digging the tunnel. [4]
\item An engineer models the height of the roof of the tunnel using the curve $y = \frac{x}{81}(108x - 54x^2 + 12x^3 - x^4)$. This curve is symmetrical about $x = 3$.
\begin{enumerate}[label=(\Alph*)]
\item Show that, according to this model, a vehicle of rectangular cross-section which is 3.6 m wide and 4.4 m high would not be able to pass through the tunnel. [2]
\item Use integration to calculate the area of the cross-section given by this model. Hence obtain another estimate of the volume of earth removed in digging the tunnel. [5]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{OCR MEI C2 2016 Q9 [11]}}