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Suppose I have two function $f(x)$ and $g(x)$ such that for $x \in (\alpha, \beta )$ we have $f(x) \ge g(x) $.

I found an "exercise solution" that state that the volume given by the rotation of the area between $f$ and $g$ is:

$$\pi\int_\alpha^\beta (f(x) - g(x))^2 dx$$

but i think it's wrong and that it should be:

$$\pi\int_\alpha^\beta f(x)^2 - g(x)^2 dx$$

Who is right?

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up vote 1 down vote accepted

I'll assume that one of the candidates is the correct solution. Now, what is the volume of a cylindrical shell of radii $R>r$ and height $1$? It is the difference $$ \pi R^2 - \pi r^2 = \pi \left(R^2-r^2 \right). $$ This does not coincide with $\pi \left( R-r \right)^2$.

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