| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2009 |
| 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 2 intervals, requiring basic substitution into a formula and evaluation of a simple function 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) Show or imply correct ordinates 1, 0.5, 0.414213 ... | B1 | |
| Use correct formula, or equivalent, with \(h = 1\) and three ordinates | M1 | |
| Obtain answer 1.21 with no errors seen | A1 | [3] |
| (ii) Justify the statement that the rule gives an over-estimate | B1 | [1] |
**(i)** Show or imply correct ordinates 1, 0.5, 0.414213 ... | B1 |
Use correct formula, or equivalent, with $h = 1$ and three ordinates | M1 |
Obtain answer 1.21 with no errors seen | A1 | [3]
**(ii)** Justify the statement that the rule gives an over-estimate | B1 | [1]3\\
\includegraphics[max width=\textwidth, alt={}, center]{b9556031-871d-4dd3-9523-e3438a41339f-2_451_775_559_683}
The diagram shows the curve $y = \frac { 1 } { 1 + \sqrt { } x }$ for values of $x$ from 0 to 2 .\\
(i) Use the trapezium rule with two intervals to estimate the value of
$$\int _ { 0 } ^ { 2 } \frac { 1 } { 1 + \sqrt { } x } \mathrm {~d} x$$
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 integral in part (i).
\hfill \mbox{\textit{CAIE P2 2009 Q3 [4]}}