17N.2.SL.TZ0.S_5

pestleMathematicsAASLPaper 217N· sl-5-5-integration-introduction-areas-between-curve-and-x-axissource ↗

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 $p$ .

[2]

The following diagram shows part of the graph of $f$ .

N17/5/MATME/SP2/ENG/TZ0/05.b

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.

[3]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

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]

Examiners’ report

[N/A]