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Those expressions worked for rotations about

**any**horizontal/vertical axis (not necessarily ##x## or ##y##) and for all functions.

Now I'm learning about areas of surfaces of revolution, but all the "formulas" or integrals online seem to be solely for rotations about one of the axes (##x## or ##y##). Is it possible to use a formula that is less specific, for instance, ##2\pi r## for the circumference? This would work for all rotations and it does not obscure the essence of deriving formulas for areas of surfaces of revolution. Also, in the "washer" (for lack of a better word) case, aren't we supposed to add the areas, unlike the volume case where we had to "subtract the volumes"? It seems very obvious to be, but I wanted to make sure since there is no reference to this particular case online.

Thank you in advance.