Find the area between the curves `y = x`and `y=x^2`.

Explanation

Given,
curve `x^(2)=y " " ` …(1)

and equation of line,
`y=x " " `…(2)
For points of intersection,
`x^(2)=x`
[ from equations (1) and (2) ]
When, `x=0,y=0` and when `x=1,y=1`
Therefore, the point of intersection of the parabola and the lines are (0, 0) and (1, 1).
` :. ` Required area
`=int_(0)^(1)(y_(2)-y_(1))dx=int_(0)^(1)(x-x^(2))dx`
`=[(x^(2))/(2)-(x^(3))/(3)]_(0)^(1)=((1)/(2)-(1)/(3))-0=(1)/(6)` sq. unit.