| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Multi-part: volume and stationary points |
| Difficulty | Challenging +1.2 Part (a) requires product rule differentiation and trigonometric manipulation to derive a given result—standard calculus technique. Part (b) involves setting up and evaluating a volume of revolution integral with trigonometric functions, requiring integration by parts. While multi-step, these are well-practiced C4 techniques with no novel insight required, making it moderately above average difficulty. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives4.08d Volumes of revolution: about x and y axes |
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\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{cb12f63c-f4d0-4eb8-b4a5-0ad12f926b1a-3_668_1172_1231_354}
\end{center}
\end{figure}
Figure 1 shows a graph of $y = x \sqrt { } \sin x , 0 < x < \pi$. The maximum point on the curve is $A$.
\begin{enumerate}[label=(\alph*)]
\item Show that the $x$-coordinate of the point $A$ satisfies the equation $2 \tan x + x = 0$.
The finite region enclosed by the curve and the $x$-axis is shaded as shown in Fig. 1.\\
A solid body $S$ is generated by rotating this region through $2 \pi$ radians about the $x$-axis.
\item Find the exact value of the volume of $S$.\\
(7)
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q5 [11]}}