So, basically, I've been trying for a while to prove that the function $f(x,y)=\frac{x-y}{(x+y)^3}$ is not integrable in $(0,1)\times(0,1)$. That is, I want to prove that the integral of it's absolute value is infinite,
$$\int_{(0,1)\times(0,1)}|f(x,y)| = +\infty$$
Now, I want to do this exactly by this method. I've already checked that the iterated integrals don't coincide, which implies what I'm trying to prove. However, this doesn't sit well with me, for the following reason: if I had a much more complicated function, I wouldn't be able to prove non-integrability in this manner, and I'd have to find either a function that bounds this one from below, or one such that I can somehow apply the limit comparison test. I would generally know how to do this in a one-variable scenario, but the multiple-variable case confuses me. That's why I'm asking for help.
Consider the triangular region $T$ defined by the vertices $(0,0),(1,0), (1,1/2).$ Then $\int_{(0,1)^2}|f|\ge \int_T |f|.$
I chose the region $T$ because here I could see $y$ is smaller than a fixed constant times $x.$ More precisely, for $(x,y)\in T,$ $0<y<x/2.$ Thus
$$f(x,y) \ge \frac{x-x/2}{(x+x)^3} = \frac{1}{16x^2}$$
for $(x,y)\in T.$ Now verify the integral of the last function over $T$ is $\infty.$