Solution:
Area bounded by the curves, $\left\{(x, y): y \geq x^{2}\right.$ and $\left.y=|x|\right\}$, is represented by the shaded region as

It can be observed that the required area is symmetrical about $y$-axis.
$\begin{aligned}
\text { Required area } &=2[\text { Area }(O C A O)-\text { Area }(O C A D O)] \\
&=2\left[\int_{0}^{\prime} x d x-\int_{0}^{1} x^{2} d x\right]
\end{aligned}$
$\begin{array}{l}
=2\left[\left[\frac{x^{2}}{2}\right]_{0}^{1}-\left[\frac{x^{3}}{3}\right]_{0}^{1}\right] \\
=2\left[\frac{1}{2}-\frac{1}{3}\right] \\
=2\left[\frac{1}{6}\right]=\frac{1}{3} \text {sq. units }
\end{array}$