| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2005 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Trapezium rule symmetry argument |
| Difficulty | Moderate -0.8 This is a straightforward trapezium rule application with only 2 intervals requiring basic substitution into a given function, followed by a simple observation about the curve's shape. The reasoning part (ii) requires only noting that the curve is approximately linear over the interval, which is visually obvious from the diagram—no deep analysis needed. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Show or imply correct decimal ordinates \(1.2755..., 1, 0.8223...\) | B1 | |
| Use correct formula, or equivalent, with \(h = 0.6\) and three ordinates | M1 | |
| Obtain correct answer \(1.23\) with no errors seen | A1 | 3 |
| [SR: if the area is calculated with one interval, or three or more, give D1 for a correct answer.] | ||
| (ii) Give an adequate justification, e.g. one trapezium over-estimates area and the other under-estimates, or errors cancel out | B1 | 1 |
**(i)** Show or imply correct decimal ordinates $1.2755..., 1, 0.8223...$ | B1 |
Use correct formula, or equivalent, with $h = 0.6$ and three ordinates | M1 |
Obtain correct answer $1.23$ with no errors seen | A1 | 3
[SR: if the area is calculated with one interval, or three or more, give D1 for a correct answer.] | |
**(ii)** Give an adequate justification, e.g. one trapezium over-estimates area and the other under-estimates, or errors cancel out | B1 | 12\\
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The diagram shows a sketch of the curve $y = \frac { 1 } { 1 + x ^ { 3 } }$ for values of $x$ from - 0.6 to 0.6 .\\
(i) Use the trapezium rule, with two intervals, to estimate the value of
$$\int _ { - 0.6 } ^ { 0.6 } \frac { 1 } { 1 + x ^ { 3 } } \mathrm {~d} x$$
giving your answer correct to 2 decimal places.\\
(ii) Explain, with reference to the diagram, why the trapezium rule may be expected to give a good approximation to the true value of the integral in this case.
\hfill \mbox{\textit{CAIE P3 2005 Q2 [4]}}