have you thought about doing the inner integral from 0 to 1 (since that's the range of y), and the outer integral from 0 to 1 (since that's the range of x?

Also, if you graph the pdf, you'll notice that it's the area of a circle...so why not find the probability of landing in a circle of radius 1 inscribed in a 1x1 box? This should be the area of that circle divided by the area of the square.