Axis of revolution

Question

How do I calculate the volume of a solid revolution when the axis of revolution is NOT the x or y axis? I thought you do \begin{equation} π∫_a^b f^2(x-c) - g^2(x-c) dx \end{equation} where y=c (a horizontal line) is the axis of revolution, but it doesn't always work. It seems like sometimes I'm supposed to do (c-x) instead, but I can't figure out why. Can anyone explain this to me?

Answer 1

I am guessing you want $$\pi \int_a^b (f(x)-c)^2 - (g(x)-c)^2 \; dx$$ instead? (You need, presumably, to check that $f(x)\geq c$, and $g(x)\geq c$ for $x \in [a,b]$.)