Find the area between the curves $y = 2 + x - {x^2}{\text{ and }}y = 2 - 3x + {x^2}$ .

$2 + x - {x^2} = 2 - 3x + {x^2}$     M1

$\Rightarrow 2{x^2} - 4x = 0$

$\Rightarrow 2x(x - 2) = 0$

$\Rightarrow x = 0,{\text{ }}x = 2$     A1A1

Note: Accept graphical solution.

Award M1 for correct graph and A1A1 for correctly labelled roots.

$\therefore {\text{A}} = \int_0^2 {\left( {(2 + x - {x^2}) - (2 - 3x + {x^2})} \right){\text{d}}x}$     (M1)

$= \int_0^2 {(4x - 2{x^2}){\text{d}}x\,\,\,\,\,{\text{or equivalent}}}$     A1

$= \left[ {2{x^2} - \frac{{2{x^3}}}{3}} \right]_0^2$     A1

$= \frac{8}{3}\left( { = 2\frac{2}{3}} \right)$     A1

[7 marks]

This was the question that gained the most correct responses. A few candidates struggled to find the limits of the integration or found a negative area.