  We show that geometric integrals of the type $\int_\Omega f\, d g^1\wedge \,
d g^2$ can be defined over a two-dimensional domain $\Omega$ when the functions
$f$, $g^1$, $g^2\colon \mathbb{R}^2\to \mathbb{R}$ are just H\"{o}lder
continuous with sufficiently large H\"{o}lder exponents and the boundary of
$\Omega$ has sufficiently small dimension, by summing over a refining sequence
of partitions the discrete Stratonovich or It\^{o} type terms. This leads to a
two-dimensional extension of the classical Young integral that coincides with
the integral introduced recently by R.~Z\"{u}st. We further show that the
Stratonovich-type summation allows to weaken the requirements on H\"{o}lder
exponents of the map $g=(g^1,g^2)$ when $f(x)=F(x,g(x))$ with $F$ sufficiently
regular. The technique relies upon an extension of the sewing lemma from Rough
paths theory to alternating functions of two-dimensional oriented simplices,
also proven in the paper.
