Area between a parabola and a line

Statement

Compute the area of the region between the parabola $y=x^2$ and the line $y=x+2$.

Solution

Intersections: $x^2=x+2\Rightarrow x^2-x-2=0\Rightarrow (x-2)(x+1)=0\Rightarrow x=-1,\ 2$. On this interval the line is above the parabola, so $A=\int_{-1}^{2}\big(x+2-x^2\big)\,dx=\Big[\frac{x^2}{2}+2x-\frac{x^3}{3}\Big]_{-1}^{2}=\Big(2+4-\tfrac83\Big)-\Big(\tfrac12-2+\tfrac13\Big)=\frac{9}{2}.$

$\boxed{A=\dfrac{9}{2}=4{.}5}$