| Exam Board | OCR MEI |
|---|---|
| Module | Paper 1 (Paper 1) |
| Session | Specimen |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Show trapezium rule gives specific value |
| Difficulty | Moderate -0.8 This is a straightforward application of the trapezium rule with all intermediate values provided in a spreadsheet format. Students only need to identify the strip width calculation, explain the trapezium rule weighting (halving end ordinates), and multiply by the strip width—all routine procedures requiring minimal problem-solving. |
| Spec | 1.09f Trapezium rule: numerical integration |
| A | B | C | D | E | |
| 1 | \(x\) | \(\sin x\) | \(y\) | ||
| 2 | 0 | 0.0000 | 0.0000 | 1.0000 | 0.5000 |
| 3 | 0.125 | 0.3927 | 0.3827 | 1.1759 | 1.1759 |
| 4 | 0.25 | 0.7854 | 0.7071 | 1.3066 | 1.3066 |
| 5 | 0.375 | 1.1781 | 0.9239 | 1.3870 | 1.3870 |
| 6 | 0.5 | 1.5708 | 1.0000 | 1.4142 | 0.7071 |
| 7 | 5.0766 | ||||
| 8 |
| Answer | Marks | Guidance |
|---|---|---|
| \(A3^*\pi\) oe | B1 [1] | Or \(0.125 \times \pi\) oe |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{1}{2}D2 + D3 + D4 + D5 + \frac{1}{2}D6\) | B1 [1] | Or equivalent expressed in words |
| Answer | Marks | Guidance |
|---|---|---|
| \(5.0766 \times 0.3927 = 1.9935...\) | M1 | Or \(5.0766 \times \frac{\pi}{8}\) |
| \(1.99\) (units²) (to 3sf) | A1 [2] |
## Question 6:
### Part (a):
$A3^*\pi$ oe | **B1** [1] | Or $0.125 \times \pi$ oe
### Part (b):
$\frac{1}{2}D2 + D3 + D4 + D5 + \frac{1}{2}D6$ | **B1** [1] | Or equivalent expressed in words
### Part (c):
$5.0766 \times 0.3927 = 1.9935...$ | **M1** | Or $5.0766 \times \frac{\pi}{8}$
$1.99$ (units²) (to 3sf) | **A1** [2] |
---6 Fig. 6 shows a partially completed spreadsheet.\\
This spreadsheet uses the trapezium rule with four strips to estimate $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { 1 + \sin x } \mathrm {~d} x$.
\begin{table}[h]
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
& A & B & C & D & E \\
\hline
1 & & $x$ & $\sin x$ & $y$ & \\
\hline
2 & 0 & 0.0000 & 0.0000 & 1.0000 & 0.5000 \\
\hline
3 & 0.125 & 0.3927 & 0.3827 & 1.1759 & 1.1759 \\
\hline
4 & 0.25 & 0.7854 & 0.7071 & 1.3066 & 1.3066 \\
\hline
5 & 0.375 & 1.1781 & 0.9239 & 1.3870 & 1.3870 \\
\hline
6 & 0.5 & 1.5708 & 1.0000 & 1.4142 & 0.7071 \\
\hline
7 & & & & & 5.0766 \\
\hline
8 & & & & & \\
\hline
\end{tabular}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{center}
\end{table}
\begin{enumerate}[label=(\alph*)]
\item Show how the value in cell B3 is calculated.
\item Show how the values in cells D2 to D6 are used to calculate the value in cell E7.
\item Complete the calculation to estimate $\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sqrt { 1 + \sin x } \mathrm {~d} x$.
Give your answer to 3 significant figures.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 1 Q6 [4]}}