area of the region bounded

The equation $y = x^2$ is an upward parabola and the $y = \sqrt{x}$ is a rightward parabola. The two has vertex at the origin and they meet at point (1, 1). The required area is dotted region in the figure below:
 

The area of the rectangular element is y dx, and y is the difference between the top end and bottom end of the strip. In equation,
$dA = y \, dx = (y_U - y_L) \, dx$
 

You sum (integrate) it up and you're good to go.