M3 June 2007 Q1
1. The rudder on a ship is modelled as a uniform plane lamina having the same shape as the region \(R\) which is enclosed between the curve with equation \(y = 2x - x^2\) and the \(x\)-axis.
(a) Show that the area of \(R\) is \(\dfrac{4}{3}\). (4)
(b) Find the coordinates of the centre of mass of the lamina. (5)
| Scheme | Marks |
|---|---|
| \(A = \displaystyle\int_0^2 \left(2x - x^2\right)\mathrm{d}x\) | M1 A1 |
| \(= \left[x^2 - \dfrac{x^3}{3}\right]_{\ldots}^{\ldots}\) | A1 |
| \(A = \left[x^2 - \dfrac{x^3}{3}\right]_0^2 = 4 - \dfrac{8}{3} = \dfrac{4}{3}\) * cso | A1 |
| (4) |
| Scheme | Marks |
|---|---|
| \(\bar{x} = 1\) (by symmetry) | B1 |
| \(\dfrac{4}{3}\bar{y} = \dfrac{1}{2}\displaystyle\int y^2\,\mathrm{d}x = \dfrac{1}{2}\displaystyle\int \left(2x - x^2\right)^2\mathrm{d}x\) | M1 |
| \(= \dfrac{1}{2}\displaystyle\int \left(4x^2 - 4x^3 + x^4\right)\mathrm{d}x\) | A1 |
| \(= \dfrac{1}{2}\left[\dfrac{4x^3}{3} - x^4 + \dfrac{x^5}{5}\right]\) | A1 |
| \(\dfrac{4}{3}\bar{y} = \dfrac{1}{2}\left[\dfrac{4x^3}{3} - x^4 + \dfrac{x^5}{5}\right]_0^2 = \dfrac{8}{15}\) | |
| \(\bar{y} = \dfrac{8}{15} \times \dfrac{3}{4} = \dfrac{2}{5}\) accept exact equivalents | A1 |
| (5) | |
| (9 marks) |