If $X$ is a compact smooth Riemannian manifold, why don't we integrate on a fundamental domain in the universal cover? [closed]

Question

Let $X$ be a compact connected Riemannian manifold. The metric gives a local volume form. The universal cover is orientable, and has a precompact subspace locally isometric (with the covering metric) to $X$. Since the integrals for orientable manifolds would coincide with this construction, why don't we just define the integral for non-orientable compact manifolds in this way? I.e. lift a function $X \rightarrow \mathbb{R}$ to a $\pi$-invariant function on the universal cover and integrate it on a fundamental domain, where $\pi=\pi_1(X)$