| Exam Board | OCR |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2007 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Over/underestimate justification with graph |
| Difficulty | Moderate -0.3 This is a straightforward trapezium rule application with standard strip width and function evaluation, followed by a conceptual question about concavity. The calculation is routine C2 content requiring only substitution into the trapezium rule formula. The reasoning part about over/under-estimate requires understanding that the curve is concave down (second derivative negative), making it slightly more demanding than pure calculation but still standard for this topic. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| \(\approx 5.94\) | M1, M1, A1, A1 4 | Attempt \(y\)-values for at least 4 of the \(x = 1, 1.5, 2, 2.5, 3\) only. Attempt to use correct trapezium rule. Obtain \(\frac{1}{4} \times [\sqrt{5} + 2(\sqrt{7} + \sqrt{9} + \sqrt{11}) + \sqrt{13}]\), or decimal equiv. Obtain 5.94 or better (answer only is 0/4). |
| (ii) This is an underestimate...... ...as the tops of the trapezia are below the curve | *B1, B1dep*B 2 | State underestimate. Correct statement or sketch. |
(i) area $= \frac{1}{2} \times \frac{1}{x} \times [\sqrt{5} + 2(\sqrt{7} + \sqrt{9} + \sqrt{11}) + \sqrt{13}]$
$\approx 0.25 \times 23.766...$
$\approx 5.94$ | M1, M1, A1, A1 4 | Attempt $y$-values for at least 4 of the $x = 1, 1.5, 2, 2.5, 3$ only. Attempt to use correct trapezium rule. Obtain $\frac{1}{4} \times [\sqrt{5} + 2(\sqrt{7} + \sqrt{9} + \sqrt{11}) + \sqrt{13}]$, or decimal equiv. Obtain 5.94 or better (answer only is 0/4).
(ii) This is an underestimate...... ...as the tops of the trapezia are below the curve | *B1, B1dep*B 2 | State underestimate. Correct statement or sketch.4\\
\includegraphics[max width=\textwidth, alt={}, center]{e429080f-8634-46bc-b451-7b13b871e518-2_543_857_1155_644}
The diagram shows the curve $\mathrm { y } = \sqrt { 4 \mathrm { X } + 1 }$.\\
(i) Use the trapezium rule, with strips of width 0.5 , to find an approximate value for the area of the region bounded by the curve $y = \sqrt { 4 x + 1 }$, the $x$-axis, and the lines $x = 1$ and $x = 3$. Give your answer correct to 3 significant figures.\\
(ii) State with a reason whether this approximation is an under-estimate or an over-estimate.
\hfill \mbox{\textit{OCR C2 2007 Q4 [6]}}