| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 11 |
| 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 standard C4 volumes of revolution question with straightforward coordinate substitution, rearranging the curve equation, and integration using the formula V = π∫x² dy. The integration requires partial fractions (a core C4 technique) and logarithms, but follows a predictable pattern. Part (d) is simple unit conversion. While it has 11 marks total, each step is routine for C4 students who have practiced volumes of revolution. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.08d Evaluate definite integrals: between limits |
\includegraphics{figure_2}
Figure 2 shows a sketch of the curve $C$ with equation $y = \frac{4}{x - 3}$, $x \neq 3$.
The points $A$ and $B$ on the curve have $x$-coordinates 3.25 and 5 respectively.
\begin{enumerate}[label=(\alph*)]
\item Write down the $y$-coordinates of $A$ and $B$.
[1]
\item Show that an equation of $C$ is $\frac{3y + 4}{y} = 0$, $y \neq 0$.
[1]
\end{enumerate}
The shaded region $R$ is bounded by $C$, the $y$-axis and the lines through $A$ and $B$ parallel to the $x$-axis. The region $R$ is rotated through 360° about the $y$-axis to form a solid shape $S$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Find the volume of $S$, giving your answer in the form $\pi(a + b \ln c)$, where $a$, $b$ and $c$ are integers.
[7]
\end{enumerate}
The solid shape $S$ is used to model a cooling tower. Given that 1 unit on each axis represents 3 metres,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item show that the volume of the tower is approximately 15500 m$^3$.
[2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q4 [11]}}