| Exam Board | OCR MEI |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 5 |
| 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 volume of revolution about the y-axis question requiring students to rearrange y = 4 - x² to get x² in terms of y, identify the correct limits (y from 0 to 4), and apply the standard formula V = π∫x²dy. While it requires multiple steps, all are routine applications of a standard technique with no conceptual challenges, making it slightly easier than average. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| When \(x = 0\), \(y = 4\) | B1 | |
| \(\Rightarrow V = \pi\int_0^4 x^2\,dy\) | M1 | must have integral, \(\pi\), \(x^2\) and \(dy\) soi |
| \(= \pi\int_0^4(4-y)\,dy\) | M1 | must have \(\pi\), their \((4-y)\), their numerical \(y\) limits |
| \(= \pi\left[4y - \frac{1}{2}y^2\right]_0^4\) | B1 | \(\left[4y - \frac{y^2}{2}\right]\) |
| \(= \pi(16 - 8) = 8\pi\) | A1 | |
| [5] |
## Question 4:
| Answer | Marks | Guidance |
|--------|-------|----------|
| When $x = 0$, $y = 4$ | B1 | |
| $\Rightarrow V = \pi\int_0^4 x^2\,dy$ | M1 | must have integral, $\pi$, $x^2$ and $dy$ soi |
| $= \pi\int_0^4(4-y)\,dy$ | M1 | must have $\pi$, their $(4-y)$, their numerical $y$ limits |
| $= \pi\left[4y - \frac{1}{2}y^2\right]_0^4$ | B1 | $\left[4y - \frac{y^2}{2}\right]$ |
| $= \pi(16 - 8) = 8\pi$ | A1 | |
| **[5]** | | |
---4 The part of the curve $y = 4 - x ^ { 2 }$ that is above the $x$-axis is rotated about the $y$-axis. This is shown in Fig. 4.
Find the volume of revolution produced, giving your answer in terms of $\pi$.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{8d786d33-c5c2-44a6-8273-7a3e43e552ef-3_534_595_1831_785}
\captionsetup{labelformat=empty}
\caption{Fig. 4}
\end{center}
\end{figure}
\hfill \mbox{\textit{OCR MEI C4 Q4 [5]}}