| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with exponential functions |
| Difficulty | Standard +0.3 This is a straightforward volume of revolution question requiring the standard formula V = π∫y² dx. The key simplification is that y² = 1 + e^(2x), which integrates directly to x + (1/2)e^(2x). While it involves an exponential function, the algebra is clean and the question explicitly asks to 'show that' a given answer, making it slightly easier than average. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(V = \int_0^1 \pi y^2\,dx = \int_0^1 \pi(1 + e^{2x})\,dx\) | M1 | must be \(\pi \times\) their \(y^2\) in terms of \(x\) |
| \(= \pi\left[x + \frac{1}{2}e^{2x}\right]_0^1\) | B1 | \(\left[x + \frac{1}{2}e^{2x}\right]\) only |
| \(= \pi(1 + \frac{1}{2}e^2 - \frac{1}{2})\) | M1 | substituting both \(x\) limits in a function of \(x\) |
| \(= \frac{1}{2}\pi(1 + e^2)\)* | E1 | www |
| [4] |
## Question 5:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $V = \int_0^1 \pi y^2\,dx = \int_0^1 \pi(1 + e^{2x})\,dx$ | M1 | must be $\pi \times$ their $y^2$ in terms of $x$ |
| $= \pi\left[x + \frac{1}{2}e^{2x}\right]_0^1$ | B1 | $\left[x + \frac{1}{2}e^{2x}\right]$ only |
| $= \pi(1 + \frac{1}{2}e^2 - \frac{1}{2})$ | M1 | substituting both $x$ limits in a function of $x$ |
| $= \frac{1}{2}\pi(1 + e^2)$* | E1 | www |
| **[4]** | | |
---5 Fig. 2 shows the curve $y = \sqrt { } 1 + \mathrm { e } ^ { 2 x }$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8d786d33-c5c2-44a6-8273-7a3e43e552ef-4_434_873_306_665}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
The region bounded by the curve, the $x$-axis, the $y$-axis and the line $x = 1$ is rotated through $360 ^ { \circ }$ about the $x$-axis.
Show that the volume of the solid of revolution produced is $\frac { 1 } { 2 } \pi \left( 1 + \mathrm { e } ^ { 2 } \right)$.
\hfill \mbox{\textit{OCR MEI C4 Q5 [4]}}