I came to this problem and I got very curious to know... intuitively I would say this integral is not finite but maybe it is. Let us consider $\mathbb{R}^2$ and only the region $R=\{(x,y)\in \mathbb{R}^2: \, |x|>1, |y|>1\}$. Is the following integral finite?
$$\int_{R} \frac{1}{(\max \{x,y\})^2} dxdy.$$
Any ideas? :) Thanks a lot :)
This is infinite. Take the region $\hat{R}=\{(x,y)\in \mathbb{R}^2: 1<y\leq x \leq n\}$, we have: $$ \int_{\hat{R}}\frac{1}{x^2}dxdy = \int_{1}^{n}\int_{y}^{n}\frac{1}{x^2}dxdy\\ =\int_{1}^{n}\left[-\frac{1}{x}\right]_{y}^{n}dy=\int_{1}^{n}\left[\frac{1}{y}-\frac{1}{n}\right]dy\\=\left[\ln y-\frac{y}{n}\right]_{1}^{n}=\ln n-1+\frac{1}{n} $$ Hence $$ \int_{R}\frac{1}{x^2}dxdy>\lim_{n \to \infty}\int_{\hat{R}}\frac{1}{x^2}dxdy = \lim_{n \to \infty}\left(\ln n-1+\frac{1}{n}\right)=+\infty $$