| Exam Board | OCR |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2007 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Rectangle bounds for definite integral |
| Difficulty | Standard +0.3 This is a straightforward application of upper and lower Riemann sums using rectangles. Students need to evaluate e^(x²) at five equally-spaced points (x = 0, 0.25, 0.5, 0.75, 1) and sum rectangle areas. The method is standard FP2 content requiring only careful arithmetic and calculator use, with no conceptual challenges beyond understanding that right endpoints give upper bounds and left endpoints give lower bounds. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Show area of rect. \(= \frac{1}{4}(e^{1/16} + e^{1/4} + e^{9/16} + e^1)\) | M1 | Or numeric equivalent |
| Show area \(= 1.7054\) | A1 | At least 3 d.p. correct |
| Explain the \(< 1.71\) in terms of areas | B1 | AG. Inequality required |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Identify areas for \(>\) sign | B1 | Inequality or diagram required |
| Show area of rect. \(= \frac{1}{4}(e^0 + e^{1/16} + e^{1/4} + e^{9/16})\) | M1 | Or numeric evidence |
| Get \(A > 1.27\) | A1 | cao; or answer which rounds down |
## Question 3:
### Part (i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Show area of rect. $= \frac{1}{4}(e^{1/16} + e^{1/4} + e^{9/16} + e^1)$ | M1 | Or numeric equivalent |
| Show area $= 1.7054$ | A1 | At least 3 d.p. correct |
| Explain the $< 1.71$ in terms of areas | B1 | AG. Inequality required |
### Part (ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| Identify areas for $>$ sign | B1 | Inequality or diagram required |
| Show area of rect. $= \frac{1}{4}(e^0 + e^{1/16} + e^{1/4} + e^{9/16})$ | M1 | Or numeric evidence |
| Get $A > 1.27$ | A1 | cao; or answer which rounds down |
---3\\
\includegraphics[max width=\textwidth, alt={}, center]{268b605f-eb86-40df-946a-210da1355e83-2_686_967_998_589}
The diagram shows the curve with equation $y = \mathrm { e } ^ { x ^ { 2 } }$, for $0 \leqslant x \leqslant 1$. The region under the curve between these limits is divided into four strips of equal width. The area of this region under the curve is $A$.\\
(i) By considering the set of rectangles indicated in the diagram, show that an upper bound for $A$ is 1.71 .\\
(ii) By considering an appropriate set of four rectangles, find a lower bound for $A$.
\hfill \mbox{\textit{OCR FP2 2007 Q3 [6]}}