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]