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.
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.
The reason your first integral didn't work out is because the x limits are not ALWAYS true for the area we are considering.
Consider the small slice of the region ABOVE the point where the smaller circle meets the y axis, and BELOW the point there the big circle meets the y axis. Is it true for that region, to say that the you enter the region through the boundary of the smaller circle and exit through the boundary of the larger circle? No, it is not. In fact, for that small portion, you enter through the y axis, and exit through the boundary of the larger circle. Your limits do not account for that small slice of the region.
Employ my method.