A2 June 2024 Paper 1 Q4
4 The equation of a curve is \(y = \dfrac{1}{\sqrt{k^2 + x^2}}\), where \(k\) is a positive constant. The region between the \(x\)-axis, the \(y\)-axis and the line \(x = k\) is rotated through \(2\pi\) radians about the \(x\)-axis.
Given that the volume of the solid of revolution formed is 1 unit3, find the exact value of \(k\). [4]
| Scheme | Marks | AO |
|---|---|---|
| \(\displaystyle [V =]\, \pi\int_0^k \frac{1}{k^2 + x^2}\,[\mathrm{d}x]\) | B1 | 1.1 |
| \(\pi\left[\frac{1}{k}\arctan\left(\frac{x}{k}\right)\right]_0^k\) | B1 | 1.1 |
| \([1 =] \frac{\pi^2}{4k}\) | B1 | 1.1 |
| \(k = \frac{\pi^2}{4}\) | B1 | 3.1a |
| [4] |
Notes
B1: Ignore limits and condone missing \(\mathrm{d}x\). Multiplication by \(\pi\) may appear later.
B1: \(\left[\dfrac{1}{k}\arctan\left(\dfrac{x}{k}\right)\right]\), ignore limits. Condone missing \(\pi\).
B1: oe single term provided arctan terms evaluated