Your change of variable is \(\displaystyle u= x^2+ y^2\), y= v. So \(\displaystyle u= x^2+v^2\), \(\displaystyle x^2= u- v^2\), and \(\displaystyle x= \pm\sqrt{u- v^2}\).
Now, your problem is just a little ambiguous. The two hyperbolas have two branches and there are two regions bounded by those hyperbolas and the lines y= 0 and y= 1/2, one in the first quadrant and one in the second quadrant. It's not clear to me whether the integration is in the first quadrant only, the second quadrant only, or both. In the first quadrant, x is positive so \(\displaystyle \frac{x}{2}= \frac{\sqrt{u- v^2}}{2}\). In the second quadrant, x is negative so \(\displaystyle \frac{x}{2}= -\frac{\sqrt{u- v^2}}{2}\).
