Consider the area shown below. the top curve (in blue) is y=cuberoot(x) and the bottom curve (in red) is y=x^3, and we have used the notation dy for delta y.

The top curve is
y = √x
The bottom curve is
y = x³

We are to find the area between the two curves, the limits are 0 and 1
A = ∫(x³ - √x)dy from 0 to 1
A = x⁴/4 - (1/2)/√x |from 0 to 1
A = 0.25 square units