| Exam Board | Edexcel |
|---|---|
| Module | C34 (Core Mathematics 3 & 4) |
| Year | 2019 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with trigonometric functions |
| Difficulty | Standard +0.8 This requires computing V = π∫₀^(π/2) (x + sin 2x)² dx, expanding to get three terms including x², x sin 2x, and sin² 2x. The x sin 2x term requires integration by parts, and sin² 2x needs a double angle identity. Multiple techniques and careful algebraic manipulation are needed to reach an exact simplified fraction, making this moderately challenging but still within standard C3/C4 scope. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\int x\sin 2x\,dx = -x\cdot\frac{1}{2}\cos 2x + \frac{1}{2}\int\cos 2x\,dx\) | M1 | \(\int x\sin 2x\,dx = \pm px\cos 2x \pm q\int\cos 2x\,dx\) |
| Correct expression (\(dx\) not required) | A1 | |
| \(\int x\sin 2x\,dx = \frac{1}{4}\sin 2x - \frac{1}{2}x\cos 2x\,(+c)\) | A1 cso | Correct integration in any form. Constant of integration not required. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \((x+\sin 2x)^2 = x^2 + 2x\sin 2x + \sin^2 2x\) | B1 | Correct (possibly unsimplified) expansion |
| \(\int\sin^2 2x\,dx = \frac{1}{2}\int(1-\cos 4x)\,dx\) | M1 | Uses \(\cos 4x = \pm1\pm2\sin^2 2x\) |
| \(\int\sin^2 2x\,dx = \frac{1}{2}x - \frac{1}{8}\sin 4x\,(+c)\) | A1 | Correct integration |
| \(\int(x+\sin 2x)^2\,dx = \frac{x^3}{3}+\frac{1}{2}\sin 2x - x\cos 2x + \frac{1}{2}x - \frac{1}{8}\sin 4x\,(+c)\) | A1ft | Follow through on part (a). Constant of integration not required. |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\text{Volume} = \pi\int(x+\sin 2x)^2\,dx\) | M1 | \(\pi\) is required but may appear later |
| \(= (\pi)\left(\frac{\pi^3}{24}+0+\frac{\pi}{2}+\frac{\pi}{4}-0-(0)\right)\) | M1 | Applies limit \(\frac{\pi}{2}\) to expression of form \(\alpha x^3 + \beta x + \) (at least one trig function). Must be exact. |
| \(= \frac{\pi^4+18\pi^2}{24}\) or \(\frac{\pi^2(\pi^2+18)}{24}\) | A1 cso | Any equivalent exact single fraction from correct integration |
## Question 9:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\int x\sin 2x\,dx = -x\cdot\frac{1}{2}\cos 2x + \frac{1}{2}\int\cos 2x\,dx$ | M1 | $\int x\sin 2x\,dx = \pm px\cos 2x \pm q\int\cos 2x\,dx$ |
| Correct expression ($dx$ not required) | A1 | |
| $\int x\sin 2x\,dx = \frac{1}{4}\sin 2x - \frac{1}{2}x\cos 2x\,(+c)$ | A1 **cso** | Correct integration in any form. Constant of integration not required. |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $(x+\sin 2x)^2 = x^2 + 2x\sin 2x + \sin^2 2x$ | B1 | Correct (possibly unsimplified) expansion |
| $\int\sin^2 2x\,dx = \frac{1}{2}\int(1-\cos 4x)\,dx$ | M1 | Uses $\cos 4x = \pm1\pm2\sin^2 2x$ |
| $\int\sin^2 2x\,dx = \frac{1}{2}x - \frac{1}{8}\sin 4x\,(+c)$ | A1 | Correct integration |
| $\int(x+\sin 2x)^2\,dx = \frac{x^3}{3}+\frac{1}{2}\sin 2x - x\cos 2x + \frac{1}{2}x - \frac{1}{8}\sin 4x\,(+c)$ | A1ft | Follow through on part (a). Constant of integration not required. |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\text{Volume} = \pi\int(x+\sin 2x)^2\,dx$ | M1 | $\pi$ is required but may appear later |
| $= (\pi)\left(\frac{\pi^3}{24}+0+\frac{\pi}{2}+\frac{\pi}{4}-0-(0)\right)$ | M1 | Applies limit $\frac{\pi}{2}$ to expression of form $\alpha x^3 + \beta x + $ (at least one trig function). Must be exact. |
| $= \frac{\pi^4+18\pi^2}{24}$ or $\frac{\pi^2(\pi^2+18)}{24}$ | A1 **cso** | Any equivalent exact single fraction from correct integration |
---9.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{ae871952-f525-44e6-8bac-09308aa1964f-34_1331_1589_264_182}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
(c) Find the exact value for the volume of this solid, giving your answer as a single, simplified fraction.
\section*{Figure 2 shows a sketch of part of the curve $C$ with equation $y = x + \sin 2 x$. \\
The region $R$, shown shaded in Figure 2, is bounded by $C$, the $x$-axis and the line with equation $x = \frac { \pi } { 2 }$ \\
The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution. \\
Figure 2 shows a sketch of part of the curve $C$ with equation $y = x + \sin 2 x$. The region $R$, shown shaded in Figure 2, is bounded by $C$, the $x$-axis and the line with equation $x = \frac { \pi } { 2 }$ The region $R$ is rotated through $2 \pi$ radians about the $x$-axis to form a solid of revolution.}
$\_\_\_\_$ simplified fraction.\\
\hfill \mbox{\textit{Edexcel C34 2019 Q9 [10]}}