| Date | May 2014 | Marks available | 3 | Reference code | 14M.2.sl.TZ2.2 |
| Level | SL only | Paper | 2 | Time zone | TZ2 |
| Command term | Find | Question number | 2 | Adapted from | N/A |
Question
Let \(f(x) = 5 - {x^2}\). Part of the graph of \(f\)is shown in the following diagram.
The graph crosses the \(x\)-axis at the points \(\rm{A}\) and \(\rm{B}\).
Find the \(x\)-coordinate of \({\text{A}}\) and of \({\text{B}}\).
The region enclosed by the graph of \(f\) and the \(x\)-axis is revolved \(360^\circ \) about the \(x\)-axis.
Find the volume of the solid formed.
Markscheme
recognizing \(f(x) = 0\) (M1)
eg \(f = 0,{\text{ }}{x^2} = 5\)
\(x = \pm 2.23606\)
\(x = \pm \sqrt 5 {\text{ (exact), }}x = \pm 2.24\) A1A1 N3
[3 marks]
attempt to substitute either limits or the function into formula
involving \({f^2}\) (M1)
eg \(\pi \int {{{\left( {5 - {x^2}} \right)}^2}{\text{d}}x,{\text{ }}\pi \int_{ - 2.24}^{2.24} {\left( {{x^4} - 10{x^2} + 25} \right)} ,{\text{ }}2\pi \int_0^{\sqrt 5 } {{f^2}} } \)
\(187.328\)
volume \(= 187\) A2 N3
[3 marks]
