I'm trying to find the rea of the region bounded by:
I'm very confused about how to find the boundaries for x and y. Anyone?
The terminals for the double integral will depend on what the order of integration is. From post #5 it appears that you're integrating first with respect to y. So the first double integral you have is correct (and this has already been confirmed by Krizalid) and will give the area of the first smaller region.
Now you need to think about the second double integral that gives the second smaller region:
where a sketch of the situation (that I assume you drew) should show you what the missing integral terminals are.