You probably realize that to get area you need to integrate. Multiply out all the factors to get a quartic polynomial, which is fairly simple to integrate.

The other task is defining the limits of integration. You want the domain between the two points where the graph of y crosses the x-axis.

Set y = 0 and solve for x.

x^2(1 − x)^2 = 0

x = 0 or x = 1

so your limits are 0 to 1.

x²(1 - x)² = 0

x = 0 and x = 1

ʃx²(1 - x²)² dx from 0 to 1

ʃx²(1 - 2x² + x^4) dx from 0 to 1

x^3/3 - 2/5*x^5 + (x^7)/7 eval. from 0 to 1

1/3 - 2/5 + 1/7 = 10/21 - 2/5 = 8/105 sq. units

roots are x = 0 and x = 1

integration limit [0,1] (x^2 - x^4) dx

1/3 - 1/5

2/15