On an orientable but not oriented manifold, there are two choices of orientation; either choice allows one to integrate -forms over compact subsets, with the two choices differing by a sign. On non-orientable manifold, -forms and densities cannot be identified —notably, any top-dimensional form must vanish somewhere (there are no volume forms on non-orientable manifolds), but there are nowhere-vanishing densities— thus while one can integrate densities over compact subsets, one cannot integrate -forms. One can instead identify densities with top-dimensional pseudoforms.