Find the area bounded by the curves $y\ge x^2$ and $y=|x|$.

The curves intersect where $x^2=|x|$.For $x\ge 0$, $x^2=x⟹x=0,1$.For $x<0$, $x^2=-x⟹x=0,-1$.The region is symmetric about the $y$ -axis.Total Area = $2\times {\int }_0^1(\text{upper curve}-\text{lower curve})dx$ The line $y=x$ is the upper curve and $y=x^2$ is the lower curve in $[0,1]$.
$$\text{Area}=2{\int }_0^1(x-x^2)dx=2{\left[\frac{x^2}{2}-\frac{x^3}{3}\right]}_0^1$$
$$\text{Area}=2\left(\frac{1}{2}-\frac{1}{3}\right)=2\left(\frac{1}{6}\right)=\frac{1}{3}\text{ sq. units}$$