Double integral, change of variables or no

[nautolian]

Homework Statement

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

Homework Equations

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!?

 

[SammyS]

Homework Statement

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

Homework Equations

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!?

e^3y+2x is neither symmetric in x nor y, so you can't take 4 times the integral over 1/4 the region.

 

[nautolian]

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

 

[SammyS]

...
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!?

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

You need to integrate of all of region S.