Find the area of the region [ (x,y)`:x^2 le y le x )`

Explanation

Consider the equations
`y=x^2 " "(i)`
and y= x … (ii)
Clearly `y=x^2` is an upward
Parabola and y= x is a line passing through (0,0)
Solving (i) and (ii) simutaneously we get
`x^2 rArr x^2 -x x(x-1) =0`
` rArr x =0 or x =1.`
From (ii) =0 `rArr y=0) "and " (x-1 rArr y=1)`
So the points of intersection of (i) and (ii) are O(0,0) and S(1,1)
Draw `AM_|_ OX`
Required are =shaded area
=(area OMAO )-(area OMACO )
`=underset(0) overset(1)int "(0 y for line )dx" -underset(0)overset(1)int"(y for parabola )"bx `
`=underset(0)overset(1)intx dx - underset(0)overset(1)intx^2dx=[(x^2)/(2)]_0^1-[(x^3)/(3)]_0^(1)=[1/2-1/3]=1/6`
Hence the required area is `1/6` sq unit.