Find the $x$-intercepts of the graph of $f$.

The region enclosed by the graph of $f$ and the $x$-axis is rotated $360^\circ$ about the $x$-axis.

Find the volume of the solid formed.

valid approach     (M1)

eg     $f(x) = 0$, sketch of parabola showing two $x$-intercepts

$x = 1,{\text{ }}x = 4{\text{   }}\left( {{\text{accept (1, 0), (4, 0)}}} \right)$     A1A1     N3

[3 marks]

attempt to substitute either limits or the function into formula involving ${f^2}$     (M1)

eg     $\int_1^4 {{{\left( {f(x)} \right)}^2}{\text{d}}x,{\text{ }}\pi \int {{{\left( {(x - 1)(x - 4)} \right)}^2}} }$

${\text{volume}} = 8.1\pi {\text{   (exact), 25.4}}$     A2     N3

[3 marks]