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.

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]