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.

Well, I hate to say it, but it all depends.

What I have is a function that depends on the vector between volume elements within two volumes. I am integrating this function over both volumes. The volumes are spheres. I can define the function or the volumes in spherical or cylindrical coordinates. The function essentially "weights" the vector for each volume element pair.

When I set the function to unity, the two integrals produce the same volumes. When I allow the function to take on a value related to the length of the vector between volume elements, the value of this integral differs as the ratio of the sphere radii is varied. Hence, I get a range of values over the range of the radii ratio. When I switch from cylindrical to spherical coordinates, the value of the integral is not the same between the two coordinate systems for a given ratio. Hence, I get a type of scatter plot, but the scatter plot is different between the two coordinate systems.