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 August 2nd, 2017, 04:34 PM #1 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 changing of variables and determining the boundaries Evaluate $\displaystyle \int \int (x+y) e^{x-y}$ over the triangle with corners (0, 0), (-1, 1), and (1, 1), using x = (u + v)/2, y = (u - v)/2 Is there any systematic method to determine the new boundaries of u and v? Last edited by zollen; August 2nd, 2017 at 05:30 PM.
 August 2nd, 2017, 05:26 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 2,315 Thanks: 1230 you lost me at $x = (u+v)=2, ~y=(u-v)=2$ do you mean $x = \dfrac{u+v}{2},~y = \dfrac{u-v}{2}$ ?
 August 2nd, 2017, 05:30 PM #3 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 Sorry, I just fixed my first post.
 August 2nd, 2017, 06:20 PM #4 Senior Member   Joined: Jan 2017 From: Toronto Posts: 209 Thanks: 3 You may have forgotten the Jacobian. No matter. You have already answered my question.
August 2nd, 2017, 06:58 PM   #5
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 Originally Posted by zollen You may have forgotten the Jacobian. No matter. You have already answered my question.
I did, which is why I deleted the post. I wasn't getting the same answer from both integrations.

But as you noticed they were off by a factor of 2 which is the determinant of the Jacobian, which in this case is just the transform matrix.

 August 2nd, 2017, 09:46 PM #6 Global Moderator   Joined: Dec 2006 Posts: 20,301 Thanks: 1971 The triangle's vertices will be given by (u, v) = (0, 0), (0, -2) and (2, 0). The u-axis and the v-axis are perpendicular. Make a rough sketch.

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