The region of integration in the xy-plane is bounded by the following four lines:

x + y = 2 when x > 0 and y > 0, that is, the line y = -x + 2 in the first quadrant.

-x + y = 2 when x < 0 and y > 0, that is, the line y = x + 2 in the second quadrant.

-x - y = 2 when x < 0 and y < 0, that is, the line y = -x - 2 in the third quadrant.

x - y = 2 when x > 0 and y < 0, that is, the line y = x - 2 in the fourth quadrant.

The region is therefore a diamond centered on the origin.

Life is much easier if you now make the following coordinate transformation (suggested by this region):

.... (1)

.... (2)

It follows from (1) and (2) that and . Therefore the Jacobian of the transformation is ........

The region of integration in the uv-plane is a square. The integrations are trivial.