Question

Find the area bounded by curves $\left\{\right(x,y):y\ge x^{2}andy=|x|\}$

Answer 1

The area bounded by curves $\left\{\right(x,y):y\ge x^{2}andy=|x|\}$, is represented by the shaded region as
It can be observed that the required area is symmetrical about $y$-axis.
Required area $=2\left[Area\right(OCAO)-Area(OCADO\left)\right]$
$=2\left[{\int }_0^{1}xdx-{\int }_0^1{x}^{2}dx\right]$
$=2[{\left[\frac{x^2}{2}\right]}_0^1-{\left[\frac{x^3}{3}\right]}_0^1]$
$=2[\frac{1}{2}-\frac{1}{3}]$
$=2\left[\frac{1}{6}\right]=\frac{1}{3}$ sq. units