Notice that the area under given conditions is nothing but the area bounded between the curves y = x^2 and y = x^4 in the first quadrant.
For this we will find the points of intersection of the two curves. that is got by solving the two equations and we get the x coordinates of their points of intersection as
0 and 1.
So the required area would be area under the curve y = x^2 - area under the curve y = x^4 from 0 and 1
= integral from 0 to 1 x^2 dx - integral from 0 to 1 x^4 = 2/15