| Exam Board | OCR |
|---|---|
| Module | H240/02 (Pure Mathematics and Statistics) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Rectangle bounds for definite integral |
| Difficulty | Moderate -0.8 This is a straightforward question on numerical integration using rectangles. Part (a) requires basic arithmetic summing rectangle areas, part (b) involves simple averaging of bounds (both give 0.79 to 2sf), and part (c) asks for the standard integral notation. All parts are routine applications of well-practiced techniques with no problem-solving or insight required. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08g Integration as limit of sum: Riemann sums |
| Width \(\delta x\) | 0.1 | 0.05 | 0.025 | 0.0125 |
| Lower bound for area \(A\) | 0.73 | 0.761 | 0.776 | 0.784 |
| Upper bound for area \(A\) | 0.855 | 0.823 | 0.807 | 0.799 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.1(1+1.1^2+1.2^2+1.3^2+1.4^2)\) and \(0.1(1.1^2+1.2^2+1.3^2+1.4^2+1.5^2)\) | B1 | NB. Check working. Both seen oe |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(0.79\) | B1 | Not \(0.7915\) |
| About half way between the last two bounds or \((0.784+0.799)\div 2=0.79\). Ignore all else | B1 | Condone "The mean of the last two bounds" or other sensible. Allow UB and LB are converging towards \(0.79\) oe. The two B1 marks are independent |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(\lim_{\delta x\to 0}\displaystyle\sum_{x=1}^{1.5} y\,\delta x\) | B1 | for \(\lim_{\delta x\to 0}\sum y\,\delta x\). Allow \(x^2\) instead of \(y\) |
| B1 | for limits, dep using \(\Sigma\) not integral. \(\lim_{\delta x\to 0}\sum_{1}^{1.5}y\,\delta x\). B1B0 |
## Question 6(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.1(1+1.1^2+1.2^2+1.3^2+1.4^2)$ and $0.1(1.1^2+1.2^2+1.3^2+1.4^2+1.5^2)$ | **B1** | NB. Check working. Both seen oe |
---
## Question 6(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $0.79$ | **B1** | Not $0.7915$ |
| About half way between the last two bounds or $(0.784+0.799)\div 2=0.79$. Ignore all else | **B1** | Condone "The mean of the last two bounds" or other sensible. Allow UB and LB are converging towards $0.79$ oe. The two B1 marks are independent |
---
## Question 6(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| $\lim_{\delta x\to 0}\displaystyle\sum_{x=1}^{1.5} y\,\delta x$ | **B1** | for $\lim_{\delta x\to 0}\sum y\,\delta x$. Allow $x^2$ instead of $y$ |
| | **B1** | for limits, dep using $\Sigma$ not integral. $\lim_{\delta x\to 0}\sum_{1}^{1.5}y\,\delta x$. B1B0 |
---6 Alex is investigating the area, $A$, under the graph of $y = x ^ { 2 }$ between $x = 1$ and $x = 1.5$. They draw the graph, together with rectangles of width $\delta x = 0.1$, and varying heights $y$.\\
\includegraphics[max width=\textwidth, alt={}, center]{7298e7b9-ad52-480c-bc2b-8289aeab9ebb-06_531_714_356_251}
\begin{enumerate}[label=(\alph*)]
\item Use the rectangles in the diagram to show that lower and upper bounds for the area $A$ are 0.73 and 0.855 respectively.
\item Alex finds lower and upper bounds for the area $A$, using widths $\delta x$ of decreasing size. The results are shown in the table. Where relevant, values are given correct to 3 significant figures.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | }
\hline
Width $\delta x$ & 0.1 & 0.05 & 0.025 & 0.0125 \\
\hline
Lower bound for area $A$ & 0.73 & 0.761 & 0.776 & 0.784 \\
\hline
Upper bound for area $A$ & 0.855 & 0.823 & 0.807 & 0.799 \\
\hline
\end{tabular}
\end{center}
Use Alex's results to estimate the value of $A$ correct to $\mathbf { 2 }$ significant figures. Give a brief justification for your estimate.
\item Write down an expression, in terms of $y$ and $\delta x$, for the exact value of the area $A$.
\end{enumerate}
\hfill \mbox{\textit{OCR H240/02 2021 Q6 [5]}}