Is this correct?
Nice try, actually, but not quite...
You are right to split it up into two pieces. You missed a couple of things.
The portion above the x-axis needs in the arguments and should be on [0,1].
The portion below the x-axis should be on [-1,0].
In every case, it should be "1-". I cannot tell where you managed any "1+". Further, if your limits are identical, you ARE going to get zero (0). Try negative on the bottom and positive on the top.
The region of integration is a full circle of radius one centered around the origin. Now think about it, if you rotate it ninety degrees so you are first going in respect to x, everything will be "the same" since a circle is completely symmetric, so all you would do is switch the x's and y's
See how important it is to write clearly and check what it is you have written?
My original post took you at your word. Check that internal integral. The bottom limit is from a circle. The top limit is from a parabola. It is for this reason that I recommended splitting it up. If you've really only circles, the original integral simply is wrong and there is no need to do any splitting.