| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with exponential functions |
| Difficulty | Standard +0.3 This is a straightforward volumes of revolution question requiring the standard formula V = π∫y² dx with y = 1/(1+e^(-2x)). While it involves an exponential function, the integration is routine using substitution (u = 1+e^(-2x)), making it slightly easier than average for a C4 question. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(V = \int \pi y^2\,dx\) | M1 | Correct formula |
| \(= \int_0^1 \pi(1 + e^{-2x})\,dx\) | M1 | \(k\int_0^1(1 + e^{-2x})\,dx\) |
| \(= \pi\left[x - \frac{1}{2}e^{-2x}\right]_0^1\) | B1 | \(\left[x - \frac{1}{2}e^{-2x}\right]\) |
| \(= \pi(1 - \frac{1}{2}e^{-2} + \frac{1}{2})\) | M1 | substituting limits; must see \(0\) used; condone omission of \(\pi\) |
| \(= \pi(1\frac{1}{2} - \frac{1}{2}e^{-2})\) | A1 | o.e. but must be exact |
| [5] |
## Question 8:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $V = \int \pi y^2\,dx$ | M1 | Correct formula |
| $= \int_0^1 \pi(1 + e^{-2x})\,dx$ | M1 | $k\int_0^1(1 + e^{-2x})\,dx$ |
| $= \pi\left[x - \frac{1}{2}e^{-2x}\right]_0^1$ | B1 | $\left[x - \frac{1}{2}e^{-2x}\right]$ |
| $= \pi(1 - \frac{1}{2}e^{-2} + \frac{1}{2})$ | M1 | substituting limits; must see $0$ used; condone omission of $\pi$ |
| $= \pi(1\frac{1}{2} - \frac{1}{2}e^{-2})$ | A1 | o.e. but must be exact |
| **[5]** | | |8 Fig. 4 shows a sketch of the region enclosed by the curve . $\mathrm { J } 1 + \mathrm { e } - 2 \mathrm { x }$, the x -axis, the y -axis and the line $\boldsymbol { x } = 1$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8d786d33-c5c2-44a6-8273-7a3e43e552ef-6_442_628_314_745}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}
Find the volume of the solid generated when this region is rotated through $360 ^ { \circ }$ about the $\boldsymbol { x }$-axis. Give your answer in an exact form.
\hfill \mbox{\textit{OCR MEI C4 Q8 [5]}}