M3 June 2006 Q1
1. A uniform solid is formed by rotating the region enclosed between the curve with equation \(y = \sqrt{x}\), the \(x\)-axis and the line \(x = 4\), through one complete revolution about the \(x\)-axis. Find the distance of the centre of mass of the solid from the origin \(O\). (5)
| Scheme | Marks |
|---|---|
| Use of \((\pi)\displaystyle\int y^2\,\mathrm{d}x \times \bar{x} = (\pi)\int xy^2\,\mathrm{d}x\) | M1 |
| \(\displaystyle\int x\,\mathrm{d}x \times \bar{x} = \int x^2\,\mathrm{d}x\) | |
| \(\left[\dfrac{1}{2}x^2\right]_{\ldots}^{\ldots} \times \bar{x} = \left[\dfrac{1}{3}x^3\right]_{\ldots}^{\ldots}\) | A1 = A1 |
| Using limits 0 and 4 \(\dfrac{16}{2} \times \bar{x} = \dfrac{64}{3}\) | M1 |
| \(\bar{x} = \dfrac{8}{3}\) | A1 |
| (5) | |
| (5 marks) |