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Calculating Volume from Revolved Area

  1. Feb 13, 2009 #1
    I remember calculating volume by revolving shapes around axis using integration in school. I have an arbitrary shape shape that has no easily attainable equation, if I know the area of that shape, can I calculate the volume?

    This is an example. I know this area, and want the volume if revolved around the y-axis:

    http://img149.imageshack.us/img149/1/shapepo1.th.png [Broken]

    -Eric
     
    Last edited by a moderator: May 4, 2017
  2. jcsd
  3. Feb 13, 2009 #2

    tiny-tim

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    Hi Eric! :smile:

    No … compare the volume of a sphere with the volume of a cylinder with the same cross-section area.

    Mathematically, volume = π∫(f(z))2dz, and area = ∫f(z)dz, and the integral of (f(z))2 isn't a fixed multiple of the integral of f(z). :smile:

    hmm … how about the surface area of the solid? :wink:
     
  4. Feb 13, 2009 #3
    OK, thanks for the response.
     
  5. Feb 13, 2009 #4

    HallsofIvy

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    There is a theorem (I can't quite remember the name right now) that the volume of a solid of rotation is equal to the area of region rotated and the circumference of the circle swept out by the centroid of the region. Of course, without knowing the formula for the boundary of the region you can't find that circumference exactly but for reasonably shaped regions you might be able to approximate it by seeing where lines from opposite points intersect.
     
  6. Feb 13, 2009 #5

    tiny-tim

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    Hi HallsofIvy! :smile:

    (i think you omitted the word "product" :wink:)

    yes, the centroid would be at a distance of ((1/2)∫(f(z))2dz)/area,

    so the volume is 2π times that times the area. :smile:
     
  7. Feb 15, 2009 #6

    HallsofIvy

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    Thanks, tiny-tim.
    And I now recall that it is "Pappus' Theorem".
     
  8. Feb 15, 2009 #7

    tiny-tim

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    good ol' Pappus of Alexandria …

    Hi HallsofIvy! :smile:

    At first, I thought, what's he talking about :rolleyes: … isn't Pappus' theorem that projective hexagon thing?

    Then I looked in http://en.wikipedia.org/wiki/Pappus'_Theorem#Theorems", and found that good ol' Pappus :smile: had two theorems …
    so you're absolutely right! :biggrin:
     
    Last edited by a moderator: Apr 24, 2017
  9. Feb 15, 2009 #8

    HallsofIvy

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    Remarkable!
     
  10. Feb 15, 2009 #9

    tiny-tim

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