The area bounded by the curves $\{(x,y):y\geq x^2$ and $y=|x|\},$ is represented as shown in the fig.

It can be said that the required area is symmetrical about y-axis.

Hence the required area can also be written as $2\times$ area bounded between the straight line y=x and parabola $y=x^2.$

Now to obtain the limit let us find the point of intersection.

Given y=x and $y=x^2$

Equating both the equation we get,

$x=x^2$

$x-x^2$=0

x(1-x)=0 $\Rightarrow$ x=0;x=1.

Hence the limits are 0 and 1.

The required area $A=2x\int_a^b[f(x)-g(x)]dx$

Here a=0,b=1.f(x)=1x and $g(x)=x^2$

$A=2\left\{\int_0^1 xdx -\int_0^1 x^2dx\right \}$

on integrating ,

$\:=2\left\{\begin{bmatrix}\frac{x^2}{2}\end{bmatrix}_0^1-\begin{bmatrix}\frac{x^3}{3}\end{bmatrix}_0^1\right \}$

on applying the limits we get ,

$A=2\left\{\frac{1^2}{2}-\frac{1^3}{3}\right \}$

$\;\;\;=2\times \frac{1}{6}=\frac{1}{3}sq.units.$

Hence the required area is $ \frac{1}{3}$sq.units.