| Exam Board | WJEC |
|---|---|
| Module | Further Unit 4 (Further Unit 4) |
| Session | Specimen |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Volumes of Revolution |
| Type | Applied context: real-world solid |
| Difficulty | Challenging +1.2 This is a solid of revolution problem requiring volume calculation using the shell/disk method with appropriate bounds. Part (a) involves standard integration with substitution (expressing x in terms of y), while part (b) requires solving an equation after setting up the integral. The algebra is moderately involved but follows a well-practiced technique for Further Maths students, making it above average but not exceptionally challenging. |
| Spec | 4.08d Volumes of revolution: about x and y axes |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Capacity} = \pi\int_1^9 x^2 dy\) | M1 | AO3 |
| \(= \pi\int_1^9 (y-1)^{2/3} dy\) | A1 | AO3 |
| \(= \pi\left[\frac{3}{5}(y-1)^{5/3}\right]_1^9\) | A1 | AO3 |
| \(= \frac{3\pi}{5}(32-0)\) | A1 | AO3 |
| \(= 60.3(1857...)\) | A1 | AO3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\text{Capacity} = \pi\int_1^a (y-1)^{2/3} dy\) | M1 | AO3 |
| \(= \pi\left[\frac{3}{5}(y-1)^{5/3}\right]_1^a\) | A1 | AO3 |
| \(= \frac{3\pi}{5}(a-1)^{5/3}\) | A1 | AO3 |
| Attempting to solve \(\frac{3\pi}{5}(a-1)^{5/3} = 25\) | M1 | AO3 |
| \(a = 5.72\) (5.71610...) | A1 | AO3 |
## Part (a)
$\text{Capacity} = \pi\int_1^9 x^2 dy$ | M1 | AO3
$= \pi\int_1^9 (y-1)^{2/3} dy$ | A1 | AO3
$= \pi\left[\frac{3}{5}(y-1)^{5/3}\right]_1^9$ | A1 | AO3
$= \frac{3\pi}{5}(32-0)$ | A1 | AO3
$= 60.3(1857...)$ | A1 | AO3
## Part (b)
$\text{Capacity} = \pi\int_1^a (y-1)^{2/3} dy$ | M1 | AO3
$= \pi\left[\frac{3}{5}(y-1)^{5/3}\right]_1^a$ | A1 | AO3
$= \frac{3\pi}{5}(a-1)^{5/3}$ | A1 | AO3
Attempting to solve $\frac{3\pi}{5}(a-1)^{5/3} = 25$ | M1 | AO3
$a = 5.72$ (5.71610...) | A1 | AO3
**Total: [10]**
---The curve $y = 1 + x^3$ is denoted by $C$.
\begin{enumerate}[label=(\alph*)]
\item A bowl is designed by rotating the arc of $C$ joining the points $(0,1)$ and $(2,9)$ through four right angles about the $y$-axis. Calculate the capacity of the bowl. [5]
\item Another bowl with capacity 25 is to be designed by rotating the arc of $C$ joining the points with $y$ coordinates 1 and $a$ through four right angles about the $y$-axis. Calculate the value of $a$. [5]
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 4 Q8 [10]}}