  We investigate the possibilities of integration on the minimal
$\mathbb{Z}_2^2$-superspace. Two definitions are taken from the works by Poncin
and Schouten and we examine their generalizations. It is shown that these
definitions impose some restrictions on the integrable functions. We then
introduce a new definition of integral, which is inspired by our previous work,
and show that the definition does not impose restrictions on the integrable
functions. An interesting feature of this definition is the emergence of a
spatial coordinate which means that the integral is defined on $\mathbb{R}^2$
despite the fact that the $(0,0)$ part of the minimal
$\mathbb{Z}_2^2$-superspace is $\mathbb{R}. $
