| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2010 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Indefinite & Definite Integrals |
| Type | Trapezium rule estimation |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule with only two intervals, requiring basic substitution into a given formula and evaluation of xe^{-x} at three points. Part (ii) tests understanding of concavity but is a standard textbook question requiring minimal insight. Significantly easier than average A-level questions. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State or imply correct ordinates \(0.27067..., 0.20521..., 0.14936...\) | B1 | |
| Use correct formula, or equivalent, correctly with \(h = 0.5\) and three ordinates | M1 | |
| Obtain answer \(0.21\) with no errors seen | A1 | [3] |
| (ii) Justify statement that the trapezium rule gives an over-estimate | B1 | [1] |
**(i)** State or imply correct ordinates $0.27067..., 0.20521..., 0.14936...$ | B1 |
Use correct formula, or equivalent, correctly with $h = 0.5$ and three ordinates | M1 |
Obtain answer $0.21$ with no errors seen | A1 | [3]
**(ii)** Justify statement that the trapezium rule gives an over-estimate | B1 | [1]2\\
\includegraphics[max width=\textwidth, alt={}, center]{f55e1431-9443-40d7-bec0-6f5c1d9fa2d7-2_531_949_431_598}
The diagram shows part of the curve $y = x \mathrm { e } ^ { - x }$. The shaded region $R$ is bounded by the curve and by the lines $x = 2 , x = 3$ and $y = 0$.\\
(i) Use the trapezium rule with two intervals to estimate the area of $R$, giving your answer correct to 2 decimal places.\\
(ii) State, with a reason, whether the trapezium rule gives an under-estimate or an over-estimate of the true value of the area of $R$.
\hfill \mbox{\textit{CAIE P2 2010 Q2 [4]}}