We analyze a Monte Carlo method using stratified sampling for approximate integration. We focus on integration of non-smooth functions: we consider the indicator function of a Jordan-measurable subset of the $s$-dimensional unit cube $I^s := [0,1)^s$. We prove a bound for the variance and show an improved convergence rate (compared to plain Monte Carlo). When the boundary of the subset is defined by a function on $I^{s-1}$, the variance is estimated by means of the variation of the function. The tightness of the previous bounds is assessed through numerical experiments in dimensions $s=2,3$ and $4$, where we compute sample variances.