I can solve a lot of volume integrals and their applications by dividing curves/ regions into infinitesimal elements and then integrating them. I have only seen the three coordinate systems being applied to find said differential elements for the most part. But I don't have a clear physical interpretation of the differential element of a solid beyond a cube/cylinder/sphere in nature. I am not convinced that these are the only shapes that need to have their own coordinate systems since there are elements that change that warrant differential volume in their own sense. Is there any physical and clear way to define a differential volume element of another solid (for example a cone or a pyramid)? And could we define differential volume elements of these shapes instead?
I hope this matter does not seem dumb and slight to you. I understand the bounds of different 3D volume elements dictate its overall integration. I was inquisitive into the matter if there was a difference in a differential element between two shapes. As a hypothetical example, the differential elements of a cone and a sphere are compared with a cone could be plotted as $\phi$ being the apex angle but the bounds are dependent on one another as $\rho$ depends on $\phi$. I would predict this dependence is not necessary for looking at a slice of volume in its own conical differential element rather fitting into the paradigm of spherical coordinates.
I see based on bot, my question is unclear. I am asking how to find a differential element of any shape that is not cube/cylinder/sphere. I keep mentioning this cone example, so I would hope someone would use it to try to match it up to their hypothesis. The coordinate system are defined with respect to a differential volume element of sphere/cylinder/cube, but I am curious about other shapes. We have seen them written as combinations of these three but the bounds needed to depend on another element in which I don't think that is the same as a differential volume slice alone.