A2 June 2020 Paper 2 Q7
7 The diagram shows part of the graph of \(y = \cos^{-1} x\)
The finite region enclosed by the graph of \(y = \cos^{-1} x\), the \(y\)-axis, the \(x\)-axis and the line \(x = 0.8\) is rotated by \(2\pi\) radians about the \(x\)-axis.
Use Simpson’s rule with five ordinates to estimate the volume of the solid formed.
Give your answer to four decimal places. [5 marks]
| Scheme | Marks | AO |
|---|---|---|
| Uses formula for volume of revolution Condone omission of \(\pi\) | M1 | 1.1a |
| Identifies and uses required \(x\)-values as 0, 0.2, 0.4, 0.6, 0.8 (PI by their \(y\), \(y^2\), \(\pi y^2\) values) | M1 | 1.1a |
| Correctly calculates values of \(y^2\) or \(\pi y^2\) | A1 | 1.1b |
| Substitutes their ordinates into Simpson’s rule with consistent \(h\) for their number of ordinates. Condone use of \(y\) rather than \(y^2\) or \(\pi y^2\) | M1 | 1.1a |
| Obtains correct answer, 3.4579 | A1 | 1.1b |
| (5 marks) |
Typical solution
\[V = \pi\int_0^{0.8} y^2\,\mathrm{d}x\]| \(x\) | 0 | 0.2 | 0.4 | 0.6 | 0.8 |
| \(y^2\) | 2.46740 | 1.87536 | 1.34393 | 0.85988 | 0.41409 |