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.

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]