I have a couple of questions about area elements and volume elements and why they are the form they are when we transform between different coordinate systems. Say we have some curve parametrized in terms of $ x$ and $ y$. So the corresponding area element if I were to evaluate a surface integral, say would be $ dA = dx\,dy$. Now transform the coordinates to polars, the corresponding dA would be $ r\,dr\,d\theta$.
But in cylindrical coordinates, the corresponding volume element is $ r \,dr\,d\theta$ .How can $ r\,dr\,d\theta $ be both an area element and a volume element?
Perhaps a more specific question, but say I was computing a surface integral in x and y. Then as above when I transform to polars, $ dA = r\,dr\,d\theta$. Now say I start the surface integral in polars. Then the area element would just be $ dr\,d\theta$. Why is this the case? (Also $dr\,d\theta$ does not have the right dimensions to be an area?)
As you can tell, I am rather confused at the moment. Any help is greatly appreciated.