| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Complete table then apply trapezium rule |
| Difficulty | Standard +0.2 This is a straightforward multi-part question requiring routine techniques: (a) calculator evaluation of sec(π/6), (b) standard trapezium rule application with given values, (c) volume of revolution using a standard integral of sec²(x/2). All parts are textbook exercises with no problem-solving or novel insight required, making it easier than average. |
| Spec | 1.05h Reciprocal trig functions: sec, cosec, cot definitions and graphs1.08d Evaluate definite integrals: between limits1.09f Trapezium rule: numerical integration4.08d Volumes of revolution: about x and y axes |
| \(x\) | 0 | \(\frac { \pi } { 6 }\) | \(\frac { \pi } { 3 }\) | \(\frac { \pi } { 2 }\) |
| \(y\) | 1 | 1.035276 | 1.414214 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(1.154701\) | B1 cao | Correct answer only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Area \(\approx\dfrac{1}{2}\times\dfrac{\pi}{6}\times\left[1+2(1.035276+\text{their } 1.154701)+1.414214\right]\) | B1; M1 | B1: Outside brackets \(\frac{1}{2}\times\frac{\pi}{6}\) or \(\frac{\pi}{12}\). M1: For structure of trapezium rule |
| \(=\dfrac{\pi}{12}\times 6.794168=1.7787\) (4 dp) | A1 | Anything that rounds to \(1.7787\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(V=\pi\displaystyle\int_0^{\frac{\pi}{2}}\left(\sec\left(\frac{x}{2}\right)\right)^2\,dx\) | B1 | For correct statement of \(\pi\int\left(\sec\frac{x}{2}\right)^2\). Ignore limits and \(dx\). Can be implied. |
| \(=\{\pi\}\left[2\tan\left(\dfrac{x}{2}\right)\right]_0^{\frac{\pi}{2}}\) | M1 A1 | M1: \(\pm\lambda\tan\left(\frac{x}{2}\right)\) from any working. A1: \(2\tan\left(\frac{x}{2}\right)\) or equivalent |
| \(=2\pi\) | A1 cao cso | \(2\pi\) from a correct solution only. Decimal answer \(6.283...\) without correct exact answer is A0. |
## Question 3:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $1.154701$ | B1 cao | Correct answer only |
---
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Area $\approx\dfrac{1}{2}\times\dfrac{\pi}{6}\times\left[1+2(1.035276+\text{their } 1.154701)+1.414214\right]$ | B1; M1 | B1: Outside brackets $\frac{1}{2}\times\frac{\pi}{6}$ or $\frac{\pi}{12}$. M1: For structure of trapezium rule |
| $=\dfrac{\pi}{12}\times 6.794168=1.7787$ (4 dp) | A1 | Anything that rounds to $1.7787$ |
---
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $V=\pi\displaystyle\int_0^{\frac{\pi}{2}}\left(\sec\left(\frac{x}{2}\right)\right)^2\,dx$ | B1 | For correct statement of $\pi\int\left(\sec\frac{x}{2}\right)^2$. Ignore limits and $dx$. Can be implied. |
| $=\{\pi\}\left[2\tan\left(\dfrac{x}{2}\right)\right]_0^{\frac{\pi}{2}}$ | M1 A1 | M1: $\pm\lambda\tan\left(\frac{x}{2}\right)$ from any working. A1: $2\tan\left(\frac{x}{2}\right)$ or equivalent |
| $=2\pi$ | A1 cao cso | $2\pi$ from a correct solution only. Decimal answer $6.283...$ without correct exact answer is A0. |3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{5c9f77f0-9f7c-4125-9da7-20fb8d79b05e-04_814_882_258_539}
\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, the line $x = \frac { \pi } { 2 }$ and the curve with equation
$$y = \sec \left( \frac { 1 } { 2 } x \right) , \quad 0 \leqslant x \leqslant \frac { \pi } { 2 }$$
The table shows corresponding values of $x$ and $y$ for $y = \sec \left( \frac { 1 } { 2 } x \right)$.
\begin{center}
\begin{tabular}{|l|l|l|l|l|}
\hline
$x$ & 0 & $\frac { \pi } { 6 }$ & $\frac { \pi } { 3 }$ & $\frac { \pi } { 2 }$ \\
\hline
$y$ & 1 & 1.035276 & & 1.414214 \\
\hline
\end{tabular}
\end{center}
\begin{enumerate}[label=(\alph*)]
\item Complete the table above giving the missing value of $y$ to 6 decimal places.
\item Using the trapezium rule, with all of the values of $y$ from the completed table, find an approximation for the area of $R$, giving your answer to 4 decimal places.
Region $R$ is rotated through $2 \pi$ radians about the $x$-axis.
\item Use calculus to find the exact volume of the solid formed.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2013 Q3 [8]}}