# Volume of a solid of revolution given two functions of $x$ and a horizontal axis, with shells?

If I have two functions of $x$, say $f(x)$ and $g(x)$, and I want to rotate $f(x)-g(x)$ around a given horizontal axis, say $y=c$, and I want to use shells, is this the correct formula?

$$2\pi \int ^{b} _{a} h^{-1}(x)(h(x)-c)dy,$$

where $h(x) = f(x)-g(x)$, assuming $f(x)$ is the upper function.

I'm just learning this stuff, and this is what I worked out writing on the fridge.

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Do you mean rotate the region bounded by the graphs of $f$ and $g$ about the line $y=c$? – David Mitra Dec 11 '11 at 19:16

I will assume you mean "rotate the region bounded by the graphs of $f$ and $g$ about the line $y=c$".

Using the shell method, you would revolve a horizontal line segment bounded by the graphs of $f$ and $g$ about $y=c$ to generate a shell.

Let's assume the left endpoint of the line segment at height $y$ is always given by $f^{-1}(y)$ and the right endpoint is always given by $g^{-1}(y)$. Also assume the region bounded by the graphs of $f$ and $G$ lies below the line $y=c$. And finally assume that the graphs of $f$ and $g$ have exactly two points of intersection: "the bottom" at $y=a$, and "the top" at $y=b$

The horizontal line segments would then range from $y=a$ to $y=b$.

The width of the horizontal line segment at height $y$ would be $g^{-1}(y)-f^{-1}(y)$.

Finally, the distance from the line segment at height $y$ to the line $y=c$ would be $c-y$.

The shell method would then give the integral $$2\pi\int_a^b \bigl( g^{-1}(y)-f^{-1}(y)\bigr)(c-y)\,dy.$$

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