double integral - converges or diverges

**zokomoko** · January 8th 2009, 06:55 AM

Hey, I've been trying to determine whether this integral converges by saying that it's smaller than e^-[(x+y)^2] over the same region. Then transforming it to polar coordinates and doing the limit I got infinitiy, meaning that the former integral is smaller or equal to infinity.. which doesn't help at all.
It seems like it converges, but I've failed at proving it.

please help..

**Opalg** · January 8th 2009, 08:52 AM

Originally Posted by zokomoko

Hey, I've been trying to determine whether this integral converges by saying that it's smaller than e^-[(x+y)^2] over the same region. Then transforming it to polar coordinates and doing the limit I got infinitiy, meaning that the former integral is smaller or equal to infinity.. which doesn't help at all.
It seems like it converges, but I've failed at proving it.

Hint: This function takes the value 1 everywhere on the line x+y=0. And it must be at least 1/2 everywhere in some strip on either side of that line.

**NonCommAlg** · January 8th 2009, 10:09 AM

the function is positive everywhere. so if $D=\{(x,y) \in \mathbb{R}^2: \ x \leq 0, \ -x \leq y \leq 1-x \},$ then using the fact that $e^a \geq 1+a,$ for all reals $a,$ we'll have:

$\int \int_{\mathbb{R}^2} e^{-(x+y)^4} \ dA > \int \int_D e^{-(x+y)^4} \ dA \geq \int_{-\infty}^0 \int_{-x}^{1-x} (1-(x+y)^4) \ dy \ dx=\infty.$ so the integral is divergent.

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