1. The problem statement, all variables and given/known data
[itex]\iint\limits_D x{\rm{d}}x{\rm{d}}y[/itex] where [itex]x = \sqrt{2y - y^2}, y = \sqrt{2x - x^2}[/itex]

2. Relevant equations

3. The attempt at a solution
I have figured out the region in question:

Spoiler: Region

But how do I get the limits of integration?

Ah, perhaps..
[tex]\int_0^1 \int_{1-\sqrt{1-y^2}}^{\sqrt{2y-y^2}} x {\rm{d}}x{\rm{d}}y[/tex]
In the inner integral I consider what X is bound by and then the 2nd integration would be just from 0 to 1 since that's what y is bound by. Not really worried about the actual calculation itself, but the most challenging bit is figuring out the bounds.

yep. those bounds look good. I get the same ones at least :)

edit: you could also do the integration in the opposite order (i.e. dy first), but I think that way is more time-consuming. and of course, in that method, the limits would look similar, but would have x instead of y.

@nuuskur: Of course, if you were "worried" about evaluating the integral, it would be wise to convert the equations to polar coordinates and set it up and work it that way.