| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Compare two trapezium rule estimates |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule with explicit strip numbers given. Students only need to substitute values into the standard formula twice and make a simple observation about over/under-estimation. The function is simple to evaluate, and no conceptual insight beyond routine procedure is required. |
| Spec | 1.09f Trapezium rule: numerical integration |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Values: \(x=1, y=0.5\); \(x=1.25, y=0.3902\); \(x=1.5, y=0.3077\); \(x=1.75, y=0.2462\); \(x=2, y=0.2\) | ||
| \(T_2=\frac{1}{2}\times 0.5(y_1+2y_{1.5}+y_2)\approx 0.3288\) | M1, A1, A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(T_4=\frac{1}{2}\times 0.25\left(y_1+2(y_{1.25}+y_{1.5}+y_{1.75})+y_2\right)\approx 0.3235\) | M1, A1, A1 | |
| The true value will be less than 0.3235; one could be reasonably confident it is accurate to 2 d.p. | B1, B1 |
## Question 6(i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| Values: $x=1, y=0.5$; $x=1.25, y=0.3902$; $x=1.5, y=0.3077$; $x=1.75, y=0.2462$; $x=2, y=0.2$ | | |
| $T_2=\frac{1}{2}\times 0.5(y_1+2y_{1.5}+y_2)\approx 0.3288$ | M1, A1, A1 | |
**Total: 3 marks**
## Question 6(ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| $T_4=\frac{1}{2}\times 0.25\left(y_1+2(y_{1.25}+y_{1.5}+y_{1.75})+y_2\right)\approx 0.3235$ | M1, A1, A1 | |
| The true value will be less than 0.3235; one could be reasonably confident it is accurate to 2 d.p. | B1, B1 | |
**Total: 5 marks**
---6 The graph shows part of the curve $y = \frac { 1 } { 1 + x ^ { 2 } }$.\\
\includegraphics[max width=\textwidth, alt={}, center]{62dbc58e-f498-483f-a9aa-05cb5aa44881-3_474_961_406_479}
Use the trapezium rule to estimate the area between the curve, the $x$-axis and the lines $x = 1$ and $x = 2$ using\\
(i) 2 strips,\\
(ii) 4 strips.
What can you conclude about the true value of the area?
\hfill \mbox{\textit{OCR MEI C4 Q6 [8]}}