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If we have a function y=f(x), we can calculate the surface area traced by that function when rotating about the x-axis as

1: [itex]S=∫dx\,2πf(x)[/itex],

which makes perfect sense to me. I am told that, if we have x=x(t) and y=y(t), the equivalent expression is

2: [itex]S=∫dt\,2πf(x)\sqrt{(dx/dt)^2+(dy/dt)^2}[/itex].

I find (2) a bit suspicious, since it seems that we are now integrating along the parametrized curve itself, rather than along the x-axis. In other words, it seems to me that (2) is equivalent to

3: [itex]S=∫dl\,2πf(x)[/itex],

which is not the same as (1).

Furthermore, in the case that our parametrization satisfies y=f(x), (2) becomes

4: [itex]S=∫dx\,2πf(x)\sqrt{1+(dy/dx)^2}[/itex],

which seems to me to brazenly contradict (1). In both (1) and (4), we are looking for the surface area traced out by rotating a curve y=f(x) about the x-axis, and yet we have two different expressions for the area.

What am I missing here? More to the point, what is it that makes (1) and (4) not contradict each other?

Thanks for the help!

-HJ Farnsworth

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# Shell method confusion

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