| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2008 |
| Session | June |
| 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 exponentials, the algebra is clean and the question explicitly asks to 'show' a given answer, making it slightly easier than average. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Lines drawn on diagram or reference to 2×2 blocks. | M1 | One (or more) block does not contain all 4 of the symbols 1, 2, 3 and 4. oe. |
Dividing the grid up into four 2×2 blocks gives a complete grid shown.
Lines drawn on diagram or reference to 2×2 blocks. | M1 | One (or more) block does not contain all 4 of the symbols 1, 2, 3 and 4. oe. | E12 Fig. 2 shows the curve $y = \sqrt { 1 + \mathrm { e } ^ { 2 x } }$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8ad99e2a-4cef-40b3-af8d-673b97536227-02_432_873_587_635}
\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 2008 Q2 [4]}}