Find the area of the region enclosed by the curves y = x^2 and y = sqrt(x).
Correct: B
The curves intersect when x^2 = sqrt(x), which implies x^4 = x, or x(x^3 - 1) = 0. The solutions are x = 0 and x = 1. The area is given by the integral of (sqrt(x) - x^2) from 0 to 1. Integral of sqrt(x) is (2/3)x^(3/2), and the integral of x^2 is (1/3)x^3. Evaluating at 1 and 0 gives (2/3)(1)^(3/2) - (1/3)(1)^3 - (0 - 0) = 2/3 - 1/3 = 1/3.