| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Rotation about y-axis, standard curve |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question requiring students to recognize the formula for rotation about the y-axis (V = π∫x² dy), substitute x = e^y from y = ln x, and integrate e^(2y) using standard exponential integration. The setup is given and the integration is routine, making this slightly easier than average. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| From graph: amplitude \(= 2\), so \(a = 2\) | B1 | |
| Wavelength \(\approx 20\), so \(2\pi b = 20\), giving \(b = \frac{10}{\pi} \approx 3.18\) | B1 | Accept equivalent correct values |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| From graph: trough length \(\approx 4\), crest length \(\approx 16\) | M1 | Reading values from graph |
| Ratio trough : crest \(\approx 4:16 = 1:4\) | A1 | |
| From text/formula: ratio should be consistent with shape parameter \(a/b\) | M1 | Comparing with theoretical ratio |
| Conclusion: consistent/not consistent with appropriate justification | A1 | Dependent on correct working |
# Question 3:
## Part (i):
| Answer | Marks | Guidance |
|--------|-------|----------|
| From graph: amplitude $= 2$, so $a = 2$ | B1 | |
| Wavelength $\approx 20$, so $2\pi b = 20$, giving $b = \frac{10}{\pi} \approx 3.18$ | B1 | Accept equivalent correct values |
## Part (ii):
| Answer | Marks | Guidance |
|--------|-------|----------|
| From graph: trough length $\approx 4$, crest length $\approx 16$ | M1 | Reading values from graph |
| Ratio trough : crest $\approx 4:16 = 1:4$ | A1 | |
| From text/formula: ratio should be consistent with shape parameter $a/b$ | M1 | Comparing with theoretical ratio |
| Conclusion: consistent/not consistent with appropriate justification | A1 | Dependent on correct working |3 Fig. 3 shows the curve $y = \ln x$ and part of the line $y = 2$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{9296c786-a42a-4aa5-b326-39adbb544cbc-02_250_550_979_753}
\captionsetup{labelformat=empty}
\caption{Fig. 3}
\end{center}
\end{figure}
The shaded region is rotated through $360 ^ { \circ }$ about the $y$-axis.\\
(i) Show that the volume of the solid of revolution formed is given by $\int _ { 0 } ^ { 2 } \pi \mathrm { e } ^ { 2 y } \mathrm {~d} y$.\\
(ii) Evaluate this, leaving your answer in an exact form.
\hfill \mbox{\textit{OCR MEI C4 2007 Q3 [6]}}