What is a Suitable Transformation for a Double Integral on a Parallelogram?

[vintwc]

Homework Statement

Let Ω ⊂ R^2 be the parallelogram with vertices at (1,0), (3,-1), (4,0) and (2,1). Evaluate ∫∫_Ω e^x dxdy.

Hint: It may be helpful to transform the integral by a suitable (affine) linear change of variables.

Homework Equations

The Attempt at a Solution

Ok here is what I have done:
From the sketch of the parallelogram, I have found the limits to be x=y+1 to (4-2y) and y=-1 to 1. With this, I am able to determine the solution of the integral which is 3/2*e^2 -1 -1/2*e^6.

Could anyone please verify this for me? Also, if my solution turns out to be right, how would I approach the hint to find the solution of this double integral? Thanks.

 

[LCKurtz]

Homework Statement

Let Ω ⊂ R^2 be the parallelogram with vertices at (1,0), (3,-1), (4,0) and (2,1). Evaluate ∫∫_Ω e^x dxdy.

Hint: It may be helpful to transform the integral by a suitable (affine) linear change of variables.

Homework Equations

The Attempt at a Solution

Ok here is what I have done:
From the sketch of the parallelogram, I have found the limits to be x=y+1 to (4-2y) and y=-1 to 1. With this, I am able to determine the solution of the integral which is 3/2*e^2 -1 -1/2*e^6.

Could anyone please verify this for me? Also, if my solution turns out to be right, how would I approach the hint to find the solution of this double integral? Thanks.

The sides of your parallogram are not parallel to the coordinate axes and you would have to break up the integrals into two parts; you can't go from y = -1 to 1 in one swoop. The sides of your parallelogram are of the form x - y = constants and x + 2y = constants. So if you want to transform your problem into one with constant limits you might want to consider the transformation:

u = x - y
v = x + 2y