| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2009 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Moderate -0.3 This is a straightforward three-part question requiring: (a) calculator evaluation of a cosine function, (b) standard trapezium rule application with given ordinates, and (c) routine integration of a cosine function. All techniques are standard C4 material with no problem-solving insight required, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration |
| \(x\) | 0 | \(\frac { 3 \pi } { 8 }\) | \(\frac { 3 \pi } { 4 }\) | \(\frac { 9 \pi } { 8 }\) | \(\frac { 3 \pi } { 2 }\) |
| \(y\) | 3 | 2.77164 | 2.12132 | 0 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(1.14805\) | B1 (1) | awrt 1.14805 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(A \approx \frac{1}{2}\times\frac{3\pi}{8}(\ldots)\) | B1 | |
| \(= \ldots\left(3+2(2.77164+2.12132+1.14805)+0\right)\) | M1 | 0 can be implied |
| \(= \frac{3\pi}{16}\left(3+2(2.77164+2.12132+1.14805)\right)\) | A1ft | ft their (a) |
| \(= \frac{3\pi}{16}\times 15.08202\ldots = 8.884\) | A1 (4) | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\int 3\cos\left(\frac{x}{3}\right)dx = \frac{3\sin\left(\frac{x}{3}\right)}{\frac{1}{3}} = 9\sin\left(\frac{x}{3}\right)\) | M1 A1 | |
| \(A = \left[9\sin\left(\frac{x}{3}\right)\right]_0^{\frac{3\pi}{2}} = 9-0 = 9\) | A1 (3) | cao |
# Question 2:
## Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $1.14805$ | B1 (1) | awrt 1.14805 |
## Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $A \approx \frac{1}{2}\times\frac{3\pi}{8}(\ldots)$ | B1 | |
| $= \ldots\left(3+2(2.77164+2.12132+1.14805)+0\right)$ | M1 | 0 can be implied |
| $= \frac{3\pi}{16}\left(3+2(2.77164+2.12132+1.14805)\right)$ | A1ft | ft their (a) |
| $= \frac{3\pi}{16}\times 15.08202\ldots = 8.884$ | A1 (4) | cao |
## Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\int 3\cos\left(\frac{x}{3}\right)dx = \frac{3\sin\left(\frac{x}{3}\right)}{\frac{1}{3}} = 9\sin\left(\frac{x}{3}\right)$ | M1 A1 | |
| $A = \left[9\sin\left(\frac{x}{3}\right)\right]_0^{\frac{3\pi}{2}} = 9-0 = 9$ | A1 (3) | cao |
---2.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{c2622c33-9436-4254-a728-10ba4703a28c-03_655_1079_207_427}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
Figure 1 shows the finite region $R$ bounded by the $x$-axis, the $y$-axis and the curve with equation $y = 3 \cos \left( \frac { x } { 3 } \right) , 0 \leqslant x \leqslant \frac { 3 \pi } { 2 }$.\\
The table shows corresponding values of $x$ and $y$ for $y = 3 \cos \left( \frac { x } { 3 } \right)$.
\begin{center}
\begin{tabular}{|l|l|l|l|l|l|}
\hline
$x$ & 0 & $\frac { 3 \pi } { 8 }$ & $\frac { 3 \pi } { 4 }$ & $\frac { 9 \pi } { 8 }$ & $\frac { 3 \pi } { 2 }$ \\
\hline
$y$ & 3 & 2.77164 & 2.12132 & & 0 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Complete the table above giving the missing value of $y$ to 5 decimal places.
\item Using the trapezium rule, with all the values of $y$ from the completed table, find an approximation for the area of $R$, giving your answer to 3 decimal places.
\item Use integration to find the exact area of $R$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2009 Q2 [8]}}