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Shell method confusion

  1. Oct 26, 2013 #1

    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
  2. jcsd
  3. Oct 27, 2013 #2


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    I haven't looked at in detail, but I think you are mixing rotation around the x axis with rotation around the y axis.
  4. Oct 27, 2013 #3
    I figured it out.

    Formula (1) is wrong, it should be [itex]S=∫2πf(x)dl[/itex], not [itex]dx[/itex], 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.


    -HJ Farnsworth
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