Find $\int_1^2 {{{\left( {f(x)} \right)}^2}{\text{d}}x}$.

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

The shaded region $R$ is enclosed by the graph of $f$, the $x$-axis and the lines $x = 1$ and $x = 2$.

Find the volume of the solid formed when $R$ is revolved ${360^ \circ }$ about the $x$-axis.

substituting for ${\left( {f(x)} \right)^2}$ (may be seen in integral)     A1

eg     ${\left( {{x^2}} \right)^2}{\text{, }}{x^4}$

correct integration, $\int {{x^4}{\text{d}}x = \frac{1}{5}{x^5}}$     (A1)

substituting limits into their integrated function and subtracting (in any order)     (M1)

eg     $\frac{{{2^5}}}{5} - \frac{1}{5}{\text{, }}\frac{1}{5}(1 - 4)$

$\int_1^2 {{{\left( {f(x)} \right)}^2}{\text{d}}x} = \frac{{31}}{5}( = 6.2)$     A1     N2

[4 marks]

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

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

$\frac{{31}}{5}\pi {\text{   }}( = 6.2\pi )$     A1     N2

[2 marks]