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

$\displaystyle \int_A{xy}~dA$

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:

$\displaystyle \int_0^2\int_{\sqrt(1-y^2)}^{\sqrt(4-y^2)}{xy}~dx~dy$

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:

$\displaystyle \int_1^2\int_0^{\pi/2}{sin\theta cos\theta r^3}~dx~dy$ 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.