Evaluate the spherical coordinate integral u=x+y , v= -2x + y;

∫ ∫ (-3x + 4y) dx dy
R

where R is the parallelogram bounded by the lines y = -x + 1, y = -x + 4, y = 2x + 2, y = 2x + 5

Rewrite the equations of the given boundary lines:

y = -x + 1  ==>  x + y = 1

y = -x + 4  ==>  x + y = 4

y = 2x + 2  ==>  -2x + y = 2

y = 2x + 5  ==>  -2x + y = 5

This tells us the parallelogram in the x-y plane corresponds to the rectangle in the u-v plane with 1 ≤ u ≤ 4 and 2 ≤ v ≤ 5.

Compute the Jacobian determinant for this change of coordinates:

Rewrite the integrand:

The integral is then