Calculate the area above y=x^2 and to the right of x=y^2 using integrals.

Answer 1

y = x^2 is 0 at x = 0, .25 at x = 1/2 and 1 at x = 1
y = +/- sqrt x is 0 at x=0, .707 at x = 1/2 and 1 at x = 1
sketch that and you will see that what you want is the integral from x = 0 to x = 1 of x^(1/2)dx minus the integral over the same interval of x^2 dx

Answer 2

The two cuves intersect at (x=0, y=0) and (x=1, y=1). The area between them is
(Integral of) x^1/2 - x^2 dx
0->1

Evaluate x^(3/2)/(3/2) - x^3/3.
at the two limits and take the difference