It depends on how you're handling it and precisely what you are doing. Quoting John M. Lee in *Introduction to Smooth Manifolds * he says: "There is no way to integrate real-valued functions in a coordinate independent way on a manifold." The proper way is to use differential forms. Then with a mapping between cylindrical and spherical space, say, you can pull-back the integrand from one space to the other. This involves exterior derivatives. Then if you also pay attention to the respective domains you should get invariant results.

What do you mean by a "distribution of values" for an integral? In each coordinate system there is a different expression for a volume element, which connects a volume element in the coordinate system to a Euclidean volume element. So the volume pieces being added up are not constant in value for a non-Cartesian coordinate system. But there total is, since the volume itself cannot depend on the coordinate system. Perhaps you could post code in a code block, or attach a notebook.

Edit: My statement above is I think incomplete. There is also the issue of translating the vector function from one coordinate system to the other. As pointed out below, this is a matter for differential geometry.