| Exam Board | WJEC |
|---|---|
| Module | Further Unit 4 (Further Unit 4) |
| Year | 2019 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Volume with trigonometric functions |
| Difficulty | Standard +0.3 This is a straightforward application of standard integration techniques with hyperbolic functions. Part (a) requires integration by parts with sinh x (a routine technique), part (b) is a volumes of revolution formula with cosh²2x (using the standard identity cosh²u = (1+cosh 2u)/2), and part (c) exploits symmetry. While hyperbolic functions are Further Maths content, these are textbook exercises requiring no problem-solving insight—just methodical application of learned techniques. |
| Spec | 4.07d Differentiate/integrate: hyperbolic functions4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| Area \(= \int_0^1 x\sinh(x) dx\) | ||
| \(= [x\cosh x]_0^1 - \int_0^1 \cosh x dx\) | M1A1 | A1 for all correct |
| \(= [x\cosh x]_0^1 - [\sinh x]_0^1\) | A1 | |
| \(= 0.368\left(\frac{1}{e}\right)\) | A1 | |
| M0 unsupported answer |
| Answer | Marks |
|---|---|
| Volume \(= \pi\int_0^1 \cosh^2(2x) dx\) | M1 |
| \(= \frac{\pi}{2}\int_0^1 (1 + \cosh 4x)dx\) | A1 |
| \(= \frac{\pi}{2}\left[x + \frac{1}{4}\sinh 4x\right]_0^1\) | A1 |
| \(= 12.3\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(24.6\) | B1 | FT (b) x 2 |
## Part a)
Area $= \int_0^1 x\sinh(x) dx$ | |
$= [x\cosh x]_0^1 - \int_0^1 \cosh x dx$ | M1A1 | A1 for all correct
$= [x\cosh x]_0^1 - [\sinh x]_0^1$ | A1 |
$= 0.368\left(\frac{1}{e}\right)$ | A1 |
| | M0 unsupported answer
## Part b)
Volume $= \pi\int_0^1 \cosh^2(2x) dx$ | M1 |
$= \frac{\pi}{2}\int_0^1 (1 + \cosh 4x)dx$ | A1 |
$= \frac{\pi}{2}\left[x + \frac{1}{4}\sinh 4x\right]_0^1$ | A1 |
$= 12.3$ | A1 |
## Part c)
$24.6$ | B1 | FT (b) x 2\begin{enumerate}[label=(\alph*)]
\item Find the area of the region enclosed by the curve $y = x\sinh x$, the $x$-axis and the lines $x = 0$ and $x = 1$. [4]
\item The region $R$ is bounded by the curve $y = \cosh 2x$, the $x$-axis and the lines $x = 0$ and $x = 1$. Find the volume of the solid generated when $R$ is rotated through four right-angles about the $x$-axis. [4]
\item Using your answer to part (b), find the total volume of the solid generated by rotating the region bounded by the curve $y = \cosh 2x$ and the lines $x = -1$ and $x = 1$. [1]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 4 2019 Q11 [9]}}