Let $S=\{(x,y,z) \in \mathbb R^3 \mid xyz=1\, x,y \in (0,1)\}$.The question is, is the area of this surface finite or not? First of all I choose a parametrization of S. Let $\sigma:(0,1) \times (0,1) \rightarrow S \quad \sigma(u,v) = \left(u,v,\displaystyle\frac{1}{uv}\right)$. Of course $\sigma$ is a valid parametrization of $S$.Then I compute the first fundamental form:
$$\sigma_u = \begin{pmatrix} 1 \\ 0 \\ -\displaystyle\frac{1}{u^2v}\end{pmatrix} \quad \sigma_v = \begin{pmatrix} 0 \\ 1 \\ -\displaystyle\frac{1}{uv^2}\end{pmatrix}$$ $$E = <\sigma_u,\sigma_u>=1+\displaystyle\frac{1}{u^4v^2}$$ $$F = <\sigma_u,\sigma_v>=\displaystyle\frac{1}{u^3v^3}$$ $$G = <\sigma_v,\sigma_v>=1+\displaystyle\frac{1}{u^2v^4}$$ So $EG-F^2=\displaystyle\frac{1}{v^4u^4}(u^2+v^2+u^4v^4)$ and then: $$A(S) = \iint_{(0,1) \times (0,1)}\frac{1}{v^2u^2}\sqrt{u^2+v^2+u^4v^4}$$ but now I'm stuck at that integral. I tried to use polar coordinates but nothing useful has come out. Thank you for your help!