(887, #19) Evaluate

where R is the region bounded by the lines 2 x - y = 1, 2 x - y = -3, 3 x + y = 1 and 3 x + y = -2.

Solution: The geometry of the problem is sketched below:

Thus, we set u = 3x + y (u = -2..1) and v = 2 x - y (v = -3..1). Furthermore

Next, the jacobian is equal to -1/5, so

Maple was used to evaluate the last integral:

lprint(int((u+v)*(2*u-3*v), u ));
lprint(int((u+v)*(2*u-3*v), u = -2 ..1));
2/3*u^3-1/2*v*u^2-3*v^2*u
6+3/2*v-9*v^2

lprint(int(int((u+v)*(2*u-3*v), u = -2 ..1), v ));
lprint(int(int((u+v)*(2*u-3*v), u = -2 ..1), v = -3 ..1));
6*v+3/4*v^2-3*v^3
-66