Shell Method Confusion: Explaining the Difference

[HJ Farnsworth]

Greetings,

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: $S=∫dx\,2πf(x)$,

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

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: $S=∫dl\,2πf(x)$,

which is not the same as (1).

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

4: $S=∫dx\,2πf(x)\sqrt{1+(dy/dx)^2}$,

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

 

[mathman]

I haven't looked at in detail, but I think you are mixing rotation around the x-axis with rotation around the y axis.

 

[HJ Farnsworth]

I figured it out.

Formula (1) is wrong, it should be $S=∫2πf(x)dl$, not $dx$, so there's actually no issue.

Also, the title of this thread should have had "surface of revolution", not "shell method" I always get all of those things mixed up.

Thanks.

-HJ Farnsworth