| Exam Board | OCR MEI |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Rectangle bounds for definite integral |
| Difficulty | Standard +0.3 This is a straightforward numerical integration question using lower and upper rectangular bounds. Parts (a)-(c) involve basic arithmetic with given coordinates, part (d) requires reading from a table and simple subtraction, and part (e) tests conceptual understanding of refinement. All steps are routine applications of standard A-level techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.08g Integration as limit of sum: Riemann sums |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | (a) | 0.2 × {0.96154 + 0.86207 + 0.73529 + |
| Answer | Marks |
|---|---|
| 0.73373… ≈ 0.7337 AG | M1 |
| A1 | 2.1 |
| 2.4 | allow M1A1 for calculation of exact values using formula in |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | (b) | 0.2 × {1 + 0.96154 + 0.86207 + 0.73529 + |
| Answer | Marks |
|---|---|
| 0.8337 correct to 4 dp | M1 |
| A1 | 1.1 |
| 1.1 | or (3.66866 - 0.5 + 1) × 0.2 |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | (c) | 0.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | (d) | 0.79162 ‒ 0.77912 |
| 0.0125 | M1 | |
| A1 | 3.1a | |
| 2.4 | if M0 allow SC1 for correct interval identified |
| Answer | Marks | Guidance |
|---|---|---|
| 14 | (e) | increase n oe |
| use rectangles of smaller width oe | B1 | 2.2a |
Question 14:
14 | (a) | 0.2 × {0.96154 + 0.86207 + 0.73529 +
0.60976 + 0.5} soi
0.73373… ≈ 0.7337 AG | M1
A1 | 2.1
2.4 | allow M1A1 for calculation of exact values using formula in
parts (a) and (b)
need to see 0.73373… as well as 0.7337 for A1
[2]
14 | (b) | 0.2 × {1 + 0.96154 + 0.86207 + 0.73529 +
0.60976}
0.8337 correct to 4 dp | M1
A1 | 1.1
1.1 | or (3.66866 - 0.5 + 1) × 0.2
[2]
14 | (c) | 0.1 | B1 | 1.1 | FT their 0.8337(32) ‒ 0.7337(32), dependent on award of M1
in part (b)
[1]
14 | (d) | 0.79162 ‒ 0.77912
0.0125 | M1
A1 | 3.1a
2.4 | if M0 allow SC1 for correct interval identified
eg 0.77912 to 0.79162
[2]
14 | (e) | increase n oe
use rectangles of smaller width oe | B1 | 2.2a | do not allow eg reduce interval
eg just ‘smaller’ rectangles – need to specify width reduction
[1]Fig. 14.1 shows the curve with equation $y = \frac{1}{1 + x^2}$, together with 5 rectangles of equal width.
\includegraphics{figure_14_1}
Fig. 14.2 shows the coordinates of the points A, B, C, D, E and F.
\includegraphics{figure_14_2}
\begin{enumerate}[label=(\alph*)]
\item Use the 5 rectangles shown in Fig. 14.1 and the information in Fig. 14.2 to show that a lower bound for $\int_0^1 \frac{1}{1 + x^2}\,dx$ is 0.7337, correct to 4 decimal places. [2]
\item Use the 5 rectangles shown in Fig. 14.1 and the information in Fig. 14.2 to calculate an upper bound for $\int_0^1 \frac{1}{1 + x^2}\,dx$ correct to 4 decimal places. [2]
\item Hence find the length of the interval in which your answers to parts (a) and (b) indicate the value of $\int_0^1 \frac{1}{1 + x^2}\,dx$ lies. [1]
\end{enumerate}
Amit uses $n$ rectangles, each of width $\frac{1}{n}$, to calculate upper and lower bounds for $\int_0^1 \frac{1}{1 + x^2}\,dx$, using different values of $n$. His results are shown in Fig. 14.3.
\includegraphics{figure_14_3}
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the length of the smallest interval in which Amit now knows $\int_0^1 \frac{1}{1 + x^2}\,dx$ lies. [2]
\item Without doing any calculation, explain how Amit could find a smaller interval which contains the value of $\int_0^1 \frac{1}{1 + x^2}\,dx$. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Paper 2 2022 Q14 [8]}}