(887, #11) Let R be the region bounded by the lines y = x, y = x - 2, y = -2x, and y = 3 - 2 x.
Evaluate

by using the transformation

Solution: The geometry of the problem is displayed below. The lines y = x and y = x -2 can be rewritten as
x - y = 0 and x - y = 2, and are the red lines. The lines y = -2x and y = 3 -2x can be rewritten as
2 x + y = 0 and 2 x + y = 3, and are shown in blue.

The given transformations for x and y are derived as follows:

Set u = x - y. Then u can be considered as a variable that runs from 0 to 2.

Eet v = 2x + y then v can be considered as a variable that runs from 0 to 3.

Finally the given transformation is obtained by solving these equations for x and y:

The jacobian is equal to 1/3:

Consequently,