Double integral, change of variables or no

1. Oct 1, 2012

nautolian

1. The problem statement, all variables and given/known data

∫∫Se2x+3ydydx where S is the region |2x|+|3y|≤ 1

2. Relevant equations

3. The attempt at a solution

So I've done this two ways and gotten two different answers and I'm not sure which is right. I used change of variables where where u=3y+2x and v=3y-2x and I got an answer of 24(e-1/e) with a jacobian of 12 and my bounds from -1 to 1 for both u and v. Then I did it in x, y coordinates and I got 4 times the given integral from x=0 to 1/2 and y=0 to (1-2x)/3 and I got a final answer of 2/3, please help me know which one is right!?
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Oct 1, 2012

SammyS

Staff Emeritus
e3y+2x is neither symmetric in x nor y, so you can't take 4 times the integral over 1/4 the region.

3. Oct 1, 2012

nautolian

So would the answer be 24(e-1/e) or is that wrong?

4. Oct 1, 2012

SammyS

Staff Emeritus
The integrand does not have the proper symmetry to do any such short cut.

You need to integrate of all of region S.