Find the area of the region bounded by the two parabolas `y=x^2`and `y^2=x`.

Explanation

Clearly `x^2=y` is an upward parabola with is vertex at (0,0) and `y^2 =x ` is right -handed parabola with its vertex also at (0,0).The shaded region shows the area bounded by these parabolas.

The given equations are
`x^2 =y`
and `y^2=x`
Using (i) in (ii) we get
`x^4=x rArr (x^3 -1)=0 rArr x =0 or x=1 `
Also `(x=0 rArr y=0 ) and (X =1 rArr y=1)`
So , the given curves intersect at O (0,0) and A(1,1)
Draw `AL_|_ OX`
Required area =(area OLABO)-(area OLACO)
= (area bounded by `x^2 =y ` from x =0 to x=1}
`= underset(0)overset(1)int sqrt(x) " dx "- underset(0)overset(1)int x^2 dx`
`=[2/3x^(3//2)]_0^1-[(x^3)/(3)]_0^1=(2/3 -1/3)=1/3` sq unit.
Hence , the required area is `1/3` sq unit