M3 June 2009 Q4
4. The finite region bounded by the \(x\)-axis, the curve \(y = \dfrac{1}{x^2}\), the line \(x = \dfrac{1}{4}\) and the line \(x = 1\), is rotated through one complete revolution about the \(x\)-axis to form a uniform solid of revolution.
(a) Show that the volume of the solid is \(21\pi\). (4)
(b) Find the coordinates of the centre of mass of the solid. (5)
| Scheme | Marks |
|---|---|
| Volume \(= \displaystyle\int_{\frac{1}{4}}^{1} \pi y^2\,dx = \int_{\frac{1}{4}}^{1} \pi\dfrac{1}{x^4}\,dx\) | M1A1 |
| \(= \left[\pi \times \dfrac{-1}{3x^3}\right]_{\frac{1}{4}}^{1}\) | A1ft |
| \(= \pi\left(\dfrac{-1}{3} + \dfrac{64}{3}\right) = 21\pi\) * | A1 |
| Scheme | Marks |
|---|---|
| \(21\pi\rho\bar{x} = \rho\displaystyle\int \pi y^2x\,dx = \rho\int \pi\dfrac{1}{x^4}x\,dx\) | M1A1 |
| \(21\pi\bar{x} = \pi\left[\dfrac{-1}{2x^2}\right]_{\frac{1}{4}}^{1}\) | A1ft |
| \(\bar{x} = \dfrac{1}{21}\left(\dfrac{-1}{2} + \dfrac{16}{2}\right) = \dfrac{5}{14}\) or awrt 0.36 | A1 |
| \(\bar{y} = 0\) by symmetry | B1 |
| (9 marks) |