- #1

themadhatter1

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## Homework Statement

Evaulate the integral making an appropriate change of variables.

[tex]\int\int_R(x+y)e^{x^2-y^2}dA[/tex] where R is the parallelogram enclosed by the lines x-2y=0, x-2y=4, 3x-y=1, 3x-y=8 .

## Homework Equations

## The Attempt at a Solution

I'm not sure what change of variables I should make. The way the region R is defined suggests that I should make the substitution u=x-2y, v=3x-y. Which maps the region r into a square s which is a simple region to integrate over. However, solving for x and y you obtain x=(1/5)(2v-u), y= (1/5)(v-3u). Changing the integral using these substitutions yields

[tex] \frac{1}{5}(3v-4u)e^{\frac{1}{25}(-8u^2+2uv+3v^2)}[/tex] which is not integrable.

Likewise, if you select a substitution which makes the integrand simple, say u=x+y and v=x-y you obtain a parallelogram as the region s which is not simple to integrate over. Am I missing something?