| Exam Board | OCR |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Numerical integration |
| Type | Simpson's rule application |
| Difficulty | Standard +0.3 This is a straightforward two-part integration question. Part (i) requires standard integration of polynomial and exponential functions with definite limits. Part (ii) applies Simpson's rule mechanically with clear instructions (two strips). Both parts are routine C3 techniques with no conceptual challenges or problem-solving required, making it slightly easier than average. |
| Spec | 1.08d Evaluate definite integrals: between limits1.08e Area between curve and x-axis: using definite integrals1.09f Trapezium rule: numerical integration4.08d Volumes of revolution: about x and y axes |
| Answer | Marks |
|---|---|
| \(\int_2^4 (2x - e^{kx}) \, dx\) | M1 A1 |
| \(= [x^2 - 2e^{kx}]_2^4\) | M1 A1 |
| \(= (16 - 2e^4) - (4 - 2e) = 12 + 2e - 2e^4\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(V = \pi \int_2^4 (2x - e^{kx})^2 \, dx\) | M1 | |
| x | 2 | 3 |
| \((2x - e^{kx})^2\) | 1.6428 | 2.3053 |
| \(I = \frac{1}{3} \times 1 \times [1.6428 + 0.3733 + 2(2.3053)] = 3.7458\) | M1 A1 | |
| \(\therefore V = 3.7458\pi = 11.8 \text{ (3sf)}\) | A1 | (9) |
## (i)
$\int_2^4 (2x - e^{kx}) \, dx$ | M1 A1 |
$= [x^2 - 2e^{kx}]_2^4$ | M1 A1 |
$= (16 - 2e^4) - (4 - 2e) = 12 + 2e - 2e^4$ | |
## (ii)
$V = \pi \int_2^4 (2x - e^{kx})^2 \, dx$ | M1 |
| x | 2 | 3 | 4 |
| $(2x - e^{kx})^2$ | 1.6428 | 2.3053 | 0.3733 |
$I = \frac{1}{3} \times 1 \times [1.6428 + 0.3733 + 2(2.3053)] = 3.7458$ | M1 A1 |
$\therefore V = 3.7458\pi = 11.8 \text{ (3sf)}$ | A1 | (9)
---\includegraphics{figure_7}
The diagram shows the curve with equation $y = 2x - e^{\frac{1}{2}x}$.
The shaded region is bounded by the curve, the $x$-axis and the lines $x = 2$ and $x = 4$.
\begin{enumerate}[label=(\roman*)]
\item Find the area of the shaded region, giving your answer in terms of e. [4]
\end{enumerate}
The shaded region is rotated through four right angles about the $x$-axis.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Using Simpson's rule with two strips, estimate the volume of the solid formed. [5]
\end{enumerate}
\hfill \mbox{\textit{OCR C3 Q7 [9]}}