Use a suitable double integral to evaluate the area between the curve y = x2 and the curvey = x where x, y ≥ 0.

The area between two curves can be found by integrating the absolute difference of the two functions over the interval where they intersect.

First, we need to find the points of intersection of the two curves y = x^2 and y = x. This can be done by setting the two equations equal to each other and solving for x:

x^2 = x x^2 - x = 0 x(x - 1) = 0

Setting each factor equal to zero gives the solutions x = 0 and x = 1. So, the two curves intersect at x = 0 and x = 1.

Next, we set up the double integral. The difference between the two functions is x - x^2. Since we are looking for the area between the curves, we take the absolute value of this difference. The limits of integration are from 0 to 1 (the points of intersection we found earlier).

The double integral is then:

∫ from 0 to 1 ∫ from 0 to (x - x^2) dy dx

This is a simple integral to evaluate. First, we integrate with respect to y, treating x as a constant. This gives:

∫ from 0 to 1 [y] from 0 to (x - x^2) dx = ∫ from 0 to 1 (x - x^2) dx

Next, we integrate with respect to x. This gives:

[x^2/2 - x^3/3] from 0 to 1 = (1/2 - 1/3) - (0 - 0) = 1/6

So, the area between the curves y = x^2 and y = x from x = 0 to x = 1 is 1/6 square units.