Manifolds, where its enough to have one chart for integration

Question

Assume a compact connected manifold $M$ is given as a subset of some $\mathbb{R}^m$. Assume we have a chart $\gamma:U \rightarrow M$ such that $M-f(U)$ (the set $M$ without $f(U)$) has zero measure in $M$ and call this property **. Then its enough to have this one chart to calculate the volume of $M$ (example: remove a point from a sphere). While there is the construction of a partition of one to calculate the volume in case one needs more charts, I think I never saw an example of a calculation where $M$ did not have the property ** and one really had to construct a partition of unity.(Simple textbook examples like, spheres, torus etc all seem to have property **) So can one classify all such manifolds having property **?