| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Compare two trapezium rule estimates |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule with explicit instructions (four strips, clear interval). Part (i) requires only substitution into the standard formula with basic calculator work. Part (ii) tests understanding of concavity but requires no calculation. Both parts are routine C4 content with no problem-solving or novel insight required. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y\) values: \(1.0655,\ 1.1696,\ 1.4142,\ 1.9283,\ 2.8964\) | B2,1,0 | table values |
| \(A \approx \frac{1}{2} \times 1\{1.0655 + 2.8964 + 2(1.1696 + 1.4142 + 1.9283)\}\) | M1 | formula |
| \(= 6.493\) | A1 [4] | 6.5 or better www |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Smaller, as the trapezium rule is an over-estimate in this case and the error is less with more strips | B1, B1 [2] |
# Question 5:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y$ values: $1.0655,\ 1.1696,\ 1.4142,\ 1.9283,\ 2.8964$ | B2,1,0 | table values |
| $A \approx \frac{1}{2} \times 1\{1.0655 + 2.8964 + 2(1.1696 + 1.4142 + 1.9283)\}$ | M1 | formula |
| $= 6.493$ | A1 [4] | 6.5 or better www |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Smaller, as the trapezium rule is an over-estimate in this case and the error is less with more strips | B1, B1 [2] | |
---5 (i) Use the trapezium rule with four strips to estimate $\int _ { - 2 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x$, showing your working.
Fig. 1 shows a sketch of $y = \sqrt { 1 + \mathrm { e } ^ { x } }$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ce44db53-2ec8-497b-a1d5-a8adf85e3929-5_533_1074_441_565}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
(ii) Suppose that the trapezium rule is used with more strips than in part (i) to estimate $\int _ { - 2 } ^ { 2 } \sqrt { 1 + \mathrm { e } ^ { x } } \mathrm {~d} x$. State, with a reason but no further calculation, whether this would give a larger or smaller estimate.\\[0pt]
[2]
\hfill \mbox{\textit{OCR MEI C4 Q5 [6]}}