| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Areas by integration |
| Type | Deduce related integral from numerical approximation |
| Difficulty | Standard +0.3 This is a straightforward application of the trapezium rule followed by a simple geometric deduction. Part (i) is routine numerical integration with given intervals. Part (ii) requires recognizing that region B's area equals (rectangle area minus region A), and understanding concavity to explain underestimation—accessible reasoning for P2 level with no novel insight required. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Show or imply correct ordinates \(1, \sqrt{2}\) or \(1.414, 3\) | B1 | |
| Use correct formula, or equivalent, with \(h = 1\) | M1 | |
| Obtain 3.41 | A1 | [3] |
| (ii) Obtain \(6 - 3.41\) and hence \(2.59\), following their answer to (i) provided less than 6 | B1√ | |
| Refer, in some form, to two line segments replacing curve and conclude with clear justification of given result that answer is an under-estimate. | B1 | [2] |
**(i)** Show or imply correct ordinates $1, \sqrt{2}$ or $1.414, 3$ | B1 |
Use correct formula, or equivalent, with $h = 1$ | M1 |
Obtain 3.41 | A1 | [3]
**(ii)** Obtain $6 - 3.41$ and hence $2.59$, following their answer to (i) provided less than 6 | B1√ |
Refer, in some form, to two line segments replacing curve and conclude with clear justification of given result that answer is an under-estimate. | B1 | [2]2\\
\includegraphics[max width=\textwidth, alt={}, center]{ee420db2-bef4-4c2b-8dd2-c8f439dd561e-2_645_750_429_699}
The diagram shows the curve $y = \sqrt { } \left( 1 + x ^ { 3 } \right)$. Region $A$ is bounded by the curve and the lines $x = 0$, $x = 2$ and $y = 0$. Region $B$ is bounded by the curve and the lines $x = 0$ and $y = 3$.\\
(i) Use the trapezium rule with two intervals to find an approximation to the area of region $A$. Give your answer correct to 2 decimal places.\\
(ii) Deduce an approximation to the area of region $B$ and explain why this approximation underestimates the true area of region $B$.
\hfill \mbox{\textit{CAIE P2 2011 Q2 [5]}}