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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}ldercontinuous with sufficiently large H\"{o}lder exponents and the boundary of$\Omega$ has sufficiently small dimension, by summing over a refining sequenceof partitions the discrete Stratonovich or It\^{o} type terms. This leads to atwo-dimensional extension of the classical Young integral that coincides withthe integral introduced recently by R.~Z\"{u}st. We further show that theStratonovich-type summation allows to weaken the requirements on H\"{o}lderexponents of the map $g=(g^1,g^2)$ when $f(x)=F(x,g(x))$ with $F$ sufficientlyregular. The technique relies upon an extension of the sewing lemma from Roughpaths theory to alternating functions of two-dimensional oriented simplices,also proven in the paper.

Integration of nonsmooth $\boldsymbol{2}$-forms: from Young to It\^{o} and Stratonovich