Area between two parabolas

We first find the points of intersection of the parabolas by solving their equations simultaneously. This gives $x^2=2x-x^2$ , or $2x^2-2x=0$ . Thus, $2x(x-1)=0$ , so $x=0$ or $x=1$ . The points of intersection are $0,0$ and $1,1$ . We see from the figure that the top and bottom boundaries are

$y_T=2x-x^2$ and $y_B=x^2$

The area of a typical rectangle is

$(y_T-y_B) \bigtriangleup x=(2x-x^2-x^2)\bigtriangleup x$

and the region lies between $x=0$ and $x=1$ . So the total area is

$A=\int_0^1 (2x - 2x^2)dx=2\int_0^1 (x - x^2)dx=\left[\dfrac{2x^2}{2}-\dfrac{2x^3}{3}\right]0 \to 1=\left(1-\dfrac{2}{3} \right)=\boxed{\dfrac{1}{3}}$