Find the area of the region `{(x,y):x^2 le y le |x| }.`

Explanation

Consider the equations
`x^2=y`
and y|x|
Clearly `x^2=y` represents an upward parabola with its vertx at O(0,0).All the points inside the parabola represent `x^(2) le y`
Also `y=|x|={(x, "when x " ge 0),(-x," when x " lt 0 ):}`
Thus , the shaded portion is the required region.
In each of the given equation , the equation remains unchanded when x is replaced by -x .
So, each of the given curves is symmetricaol about the y-axies.
`because ` required area =2(area OEAO).
`x^2=y`
and y=x
Using (iv) in (iii) ,we get `x^2=x rArr x(x-1)=0 rArr x =0 or x =1`
Now (`x =0 rArr y =0 ) and (x=1 rArr y= 1)`
Thus , the line y=x and the curve `x^2=y` intersect O(0,0) and A(1,1).Draw `AD _|_ OX and BC _|_ OX`
` therefore` required area
=2(area OEAO)=2[area ODAO)-(area ODAEO)]
=2 [(area between y=x and the x-axis form x=0 to x=1)-(area between x^2 =y and the x-axis from x =0 to x =1 )
`=2 underset(0)overset(1)int" y dx for the line OA-"underset(0)overset(1)int "y dx for the curve OEA"}`
`2[underset(0)overset(1)intxdx - underset(0)overset(1)intx^2dx]=2{[x^2/2]_0^1-[x^3/3]_0^1}=2[1/2-1/3]`
`=(2xx1/6)=1/3` sq unit
Hence , the required aera is `1/3` sq units