Calculate area of the surface: $$z = kxy/(1-x)$$ (https://www.geogebra.org/3d/uca6ehtc)
It's domain is restricted: $0\leq x\leq 1, 0\leq y\leq 1-x$
$k$ is just a parameter controlling its streching. I need a formula or maybe an integral to calculate the area of the surface.
You can use the following formula to calculate the area of a surface: $$\iint_{S} dS$$ And you can find $dS$ as follow: $$dS = \sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2} dA$$
So you end up with the following formula: $$ \iint_{S} dS = \iint_{D} \sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2} dA $$
And for your problem it's: $$Area \space of \space z = \iint_{S} dS = \int_{0}^{1}\int_{0}^{1-x} \sqrt{1+\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2} dydx $$
You can read this also for more explanations: MIT18_02SC_MNotes_v9.3to4