Region A is the area between the unit circle and the circle of radius 2, in the top right quadrant only. Find

Ok so that's the question. We're doing the theory behind changing variables from x,y to u,v or r,θ.

In standard x,y coordinates:

which comes out as 3. I'd write it all out but I'm not fluent in Latex and it would take hours.

In r,θ:

which comes out as 15/8. I'm pretty sure the second one is the one intended to be correct, but not sure what I'm missing. Perhaps because the x,y integral doesn't limit to the top right quadrant? I just can't work it out.