| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with implicit or parametric curves |
| Difficulty | Challenging +1.2 This is a standard volumes of revolution question with parametric equations requiring multiple techniques (converting limits, finding dx/dθ, using double angle formulas, integrating standard trig functions). Part (a) involves algebraic manipulation and showing a given result, while part (b) requires routine integration of polynomial and trig functions. The question is moderately harder than average due to the parametric setup and algebraic complexity, but follows predictable C4 patterns without requiring novel insight. |
| Spec | 1.05k Further identities: sec^2=1+tan^2 and cosec^2=1+cot^21.08c Integrate e^(kx), 1/x, sin(kx), cos(kx)4.08d Volumes of revolution: about x and y axes |
\includegraphics{figure_2}
Figure 2 shows a sketch of part of the curve $C$ with parametric equations
$$x = \tan \theta, \quad y = 1 + 2\cos 2\theta, \quad 0 \leq \theta < \frac{\pi}{2}$$
The curve $C$ crosses the $x$-axis at $(\sqrt{3}, 0)$. The finite shaded region $S$ shown in Figure 2 is bounded by $C$, the line $x = 1$ and the $x$-axis. This shaded region is rotated through $2\pi$ radians about the $x$-axis to form a solid of revolution.
\begin{enumerate}[label=(\alph*)]
\item Show that the volume of the solid of revolution formed is given by the integral
$$k \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} (16 \cos^2 \theta - 8 + \sec^2 \theta) \, d\theta$$
where $k$ is a constant.
[5]
\item Hence, use integration to find the exact value for this volume.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2013 Q7 [10]}}