| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | January |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Compare two trapezium rule estimates |
| Difficulty | Moderate -0.8 This is a straightforward trapezium rule application with standard strip width calculation (h=0.5) and function evaluations, followed by a conceptual question about concavity that requires understanding that the trapezium rule overestimates for convex curves. Both parts are routine C4 content requiring minimal problem-solving. |
| Spec | 1.09f Trapezium rule: numerical integration |
Fig. 2 shows part of the curve $y = 1 + x^3$.
(i) Use the trapezium rule with 4 strips to estimate $\int_0^2 (1 + x^3) \, dx$, giving your answer correct to 3 significant figures. [3]
(ii) Chris and Dave each estimate the value of this integral using the trapezium rule with 8 strips. Chris gets a result of 3.25, and Dave gets 3.30. One of these results is correct. Without performing the calculation, state with a reason which is correct. [2]2 Fig. 2 shows part of the curve $y = \sqrt { 1 + x ^ { 3 } }$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{5dcd4f44-4c61-4384-be1b-a8d63cb6b5aa-2_540_648_662_712}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
(i) Use the trapezium rule with 4 strips to estimate $\int _ { 0 } ^ { 2 } \sqrt { 1 + x ^ { 3 } } \mathrm {~d} x$, giving your answer correct to 3 significant figures.\\
(ii) Chris and Dave each estimate the value of this integral using the trapezium rule with 8 strips. Chris gets a result of 3.25, and Dave gets 3.30. One of these results is correct. Without performing the calculation, state with a reason which is correct.\\[0pt]
[2]
\hfill \mbox{\textit{OCR MEI C4 2007 Q2 [5]}}