# 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.