Do you know that \(\displaystyle x^2+ y^2= 1\) is the equation of a circle with center at (0, 0) and radius 1? So that \(\displaystyle x^2+ y^2\le 1\) is that circle and its interior- the disk with center at (0,0) and radius 1. If you choose to use polar coordinates, as Subhotosh Kahn suggests (and so do I), r will go from 0 to 1 and \(\displaystyle \theta\) from 0 to \(\displaystyle 2\pi\).
In fact, since you are just integrating "dA" over that disk, the integral is just its Area. What is the area of a disk with radius 1?
