| Date | November 2017 | Marks available | 3 | Reference code | 17N.2.sl.TZ0.5 |
| Level | SL only | Paper | 2 | Time zone | TZ0 |
| Command term | Find | Question number | 5 | Adapted from | N/A |
Question
Let \(f(x) = 6 - \ln ({x^2} + 2)\), for \(x \in \mathbb{R}\). The graph of \(f\) passes through the point \((p,{\text{ }}4)\), where \(p > 0\).
Find the value of \(p\).
The following diagram shows part of the graph of \(f\).
The region enclosed by the graph of \(f\), the \(x\)-axis and the lines \(x = - p\) and \(x = p\) is rotated 360° about the \(x\)-axis. Find the volume of the solid formed.
Markscheme
valid approach (M1)
eg\(\,\,\,\,\,\)\(f(p) = 4\), intersection with \(y = 4,{\text{ }} \pm 2.32\)
2.32143
\(p = \sqrt {{{\text{e}}^2} - 2} \) (exact), 2.32 A1 N2
[2 marks]
attempt to substitute either their limits or the function into volume formula (must involve \({f^2}\), accept reversed limits and absence of \(\pi \) and/or \({\text{d}}x\), but do not accept any other errors) (M1)
eg\(\,\,\,\,\,\)\(\int_{ - 2.32}^{2.32} {{f^2},{\text{ }}\pi \int {{{\left( {6 - \ln ({x^2} + 2)} \right)}^2}{\text{d}}x,{\text{ 105.675}}} } \)
331.989
\({\text{volume}} = 332\) A2 N3
[3 marks]
