`y=x^(3)-x=x(x-1)(x+1)` is a cubic polynomial function intersecting the x-axis at `(-1,0),(0,0),(1,0).`
`y=x^(2)+x=x(x+1)` is a quadratic function which is concave upward and intersect x-axis at `(-1, 0) (0,0).`
Solving curves, we get
`x^(3)-x=x^(2)+x`
`therefore" "x=-1,0,2`
The graphs of functions are as shown in the figure.
From the figure,
`"Required area "=overset(0)underset(-1)int((x^(3)-x)-(x^(2)+x))dx+overset(2)underset(0)int(x^(2)+x-(x^(3)-x))dx`
`=overset(0)underset(-1)int(x^(3)-x^(2)-2x)dx+overset(2)underset(0)int(x^(2)+2x-x^(3))dx`
`=[(x^(4))/(4)-(x^(3))/(3)-x^(2)]_(-1)^(0)+[(x^(3))/(3)+x^(2)-(x^(4))/(4)]_(0)^(2)`
`=[0-((1)/(4)+(1)/(3)-1)]+[(8)/(3)+4-4]`
`=(37)/(12)` sq. units
