| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| 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 (i) set up the integral for rotation about the y-axis using x = e^y, and (ii) integrate e^(2y) which is a standard exponential integral. Both parts are routine applications of C4 techniques with no problem-solving insight required, making it slightly easier than average. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(y = \ln x \Rightarrow x = e^y\) | B1 | |
| \(\Rightarrow V = \int_0^2 \pi x^2\,dy\) | M1 | |
| \(= \int_0^2 \pi(e^y)^2\,dy = \int_0^2 \pi e^{2y}\,dy\)* | E1 | |
| [3] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\int_0^2 \pi e^{2y}\,dy = \pi\left[\frac{1}{2}e^{2y}\right]_0^2\) | B1 | \(\frac{1}{2}e^{2y}\) |
| \(= \frac{1}{2}\pi(e^4 - 1)\) | M1 | substituting limits in \(k\pi e^{2y}\) |
| A1 | or equivalent, but must be exact and evaluate \(e^0\) as 1 | |
| [3] |
## Question 6:
### Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $y = \ln x \Rightarrow x = e^y$ | B1 | |
| $\Rightarrow V = \int_0^2 \pi x^2\,dy$ | M1 | |
| $= \int_0^2 \pi(e^y)^2\,dy = \int_0^2 \pi e^{2y}\,dy$* | E1 | |
| **[3]** | | |
### Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\int_0^2 \pi e^{2y}\,dy = \pi\left[\frac{1}{2}e^{2y}\right]_0^2$ | B1 | $\frac{1}{2}e^{2y}$ |
| $= \frac{1}{2}\pi(e^4 - 1)$ | M1 | substituting limits in $k\pi e^{2y}$ |
| | A1 | or equivalent, but must be exact and evaluate $e^0$ as 1 |
| **[3]** | | |
---6 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]{8d786d33-c5c2-44a6-8273-7a3e43e552ef-4_244_548_1346_768}
\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 Q6 [6]}}