| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.8 This is a straightforward application of standard C4 techniques: substituting values into a given function (requiring only calculator work) and applying the trapezium rule formula with provided ordinates. No integration, problem-solving, or conceptual insight is required—purely procedural execution of a memorized method. |
| Spec | 1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | \(\frac { \pi } { 4 }\) | \(\frac { \pi } { 2 }\) | \(\frac { 3 \pi } { 4 }\) | \(\pi\) |
| \(y\) | 0 | 8.87207 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y = 1.844321332...\) when \(x = \frac{\pi}{4}\) | B1 | awrt 1.84432 |
| \(y = 4.810477381...\) when \(x = \frac{\pi}{2}\) | B1 | awrt 4.81048 or 4.81047 |
| \(y = 0\) at \(x = 0\) and \(x = \pi\) (0 can be implied) | [2 marks total] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Area \(\approx \frac{1}{2} \times \frac{\pi}{4}; \times \{0 + 2(1.84432 + 4.81048 + 8.87207) + 0\}\) | B1 | Outside brackets: awrt 0.39 or \(\frac{1}{2} \times\) awrt 0.79; \(\frac{1}{2} \times \frac{\pi}{4}\) or \(\frac{\pi}{8}\) |
| Correct trapezium rule structure | M1\(\sqrt{}\) | For structure of trapezium rule \(\{\ldots\}\) |
| Correct expression inside brackets (all terms multiplied by outside constant) | A1\(\sqrt{}\) | Correct expression inside brackets which all must be multiplied by their "outside constant" |
| \(= \frac{\pi}{8} \times 31.05374... = 12.19477518... = \underline{12.1948}\) (4dp) | A1 cao | 12.1948 [4 marks total] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Area \(\approx \frac{\pi}{4} \times \left\{\frac{0+1.84432}{2} + \frac{1.84432+4.81048}{2} + \frac{4.81048+8.87207}{2} + \frac{8.87207+0}{2}\right\}\) | B1 | \(\frac{\pi}{4}\) (or awrt 0.79) and a divisor of 2 on all terms inside brackets |
| Which is equivalent to: Area \(\approx \frac{1}{2} \times \frac{\pi}{4}; \times \{0 + 2(1.84432 + 4.81048 + 8.87207) + 0\}\) | M1\(\sqrt{}\) | One of first and last ordinates, two of the middle ordinates inside brackets ignoring the 2 |
| A1\(\sqrt{}\) | Correct expression inside brackets if \(\frac{1}{2}\) was to be factorised out | |
| \(= \frac{\pi}{4} \times 15.52687... = 12.19477518... = \underline{12.1948}\) (4dp) | A1 cao | 12.1948 [4 marks total] |
# Question 1:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 1.844321332...$ when $x = \frac{\pi}{4}$ | B1 | awrt 1.84432 |
| $y = 4.810477381...$ when $x = \frac{\pi}{2}$ | B1 | awrt 4.81048 or 4.81047 |
| $y = 0$ at $x = 0$ and $x = \pi$ (0 can be implied) | | [2 marks total] |
## Part (b) – Way 1
| Answer/Working | Marks | Guidance |
|---|---|---|
| Area $\approx \frac{1}{2} \times \frac{\pi}{4}; \times \{0 + 2(1.84432 + 4.81048 + 8.87207) + 0\}$ | B1 | Outside brackets: awrt 0.39 or $\frac{1}{2} \times$ awrt 0.79; $\frac{1}{2} \times \frac{\pi}{4}$ or $\frac{\pi}{8}$ |
| Correct trapezium rule structure | M1$\sqrt{}$ | For structure of trapezium rule $\{\ldots\}$ |
| Correct expression inside brackets (all terms multiplied by outside constant) | A1$\sqrt{}$ | Correct expression inside brackets which all must be multiplied by their "outside constant" |
| $= \frac{\pi}{8} \times 31.05374... = 12.19477518... = \underline{12.1948}$ (4dp) | A1 cao | 12.1948 [4 marks total] |
## Part (b) – Way 2 (Aliter)
| Answer/Working | Marks | Guidance |
|---|---|---|
| Area $\approx \frac{\pi}{4} \times \left\{\frac{0+1.84432}{2} + \frac{1.84432+4.81048}{2} + \frac{4.81048+8.87207}{2} + \frac{8.87207+0}{2}\right\}$ | B1 | $\frac{\pi}{4}$ (or awrt 0.79) and a divisor of 2 on all terms inside brackets |
| Which is equivalent to: Area $\approx \frac{1}{2} \times \frac{\pi}{4}; \times \{0 + 2(1.84432 + 4.81048 + 8.87207) + 0\}$ | M1$\sqrt{}$ | One of first and last ordinates, two of the middle ordinates inside brackets ignoring the 2 |
| | A1$\sqrt{}$ | Correct expression inside brackets if $\frac{1}{2}$ was to be factorised out |
| $= \frac{\pi}{4} \times 15.52687... = 12.19477518... = \underline{12.1948}$ (4dp) | A1 cao | 12.1948 [4 marks total] |
> **Note:** An expression like Area $\approx \frac{1}{2} \times \frac{\pi}{4} + 2(1.84432 + 4.81048 + 8.87207)$ would score B1M1A0A0
---1.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ac7d862f-d10d-45ed-9077-ae4c7413cbf6-02_390_675_246_630}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
The curve shown in Figure 1 has equation $y = \mathrm { e } ^ { x } \sqrt { } ( \sin x ) , 0 \leqslant x \leqslant \pi$. The finite region $R$ bounded by the curve and the $x$-axis is shown shaded in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Complete the table below with the values of $y$ corresponding to $x = \frac { \pi } { 4 }$ and $\frac { \pi } { 2 }$, giving your answers to 5 decimal places.
\begin{center}
\begin{tabular}{ | c | c | c | c | c | c | }
\hline
$x$ & 0 & $\frac { \pi } { 4 }$ & $\frac { \pi } { 2 }$ & $\frac { 3 \pi } { 4 }$ & $\pi$ \\
\hline
$y$ & 0 & & & 8.87207 & 0 \\
\hline
\end{tabular}
\end{center}
\item Use the trapezium rule, with all the values in the completed table, to obtain an estimate for the area of the region $R$. Give your answer to 4 decimal places.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2008 Q1 [6]}}