# double integral - converges or diverges

• Jan 8th 2009, 07:55 AM
zokomoko
double integral - converges or diverges
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.

• Jan 8th 2009, 09:52 AM
Opalg
Quote:

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.
• Jan 8th 2009, 11:09 AM
NonCommAlg
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.