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Work versus volume

  1. Sep 23, 2008 #1

    mathwonk

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    A few years ago I challenged my class to use the method of volumes by slicing to
    compute the volume of a 4 ball,

    knowing the volume of a 3-ball. This leads to a slightly challenging trig
    integral, but not out of reach for a strong

    calculus student. (no one did it however.)

    This Fall while teaching the concept of work, I noticed the work integral for
    pumping liquid from a tank,

    is just the "cylindrical shells" integral for 4 dimensional volume (except for
    the factor of 2 pi).

    So if they have computed the relatively easy integral of work

    to empty a unit radius hemispherical tank of unit density liquid as pi/4,

    it follows that the volume of a 4 ball of radius a, is (2pi)(pi/4)a^4 = (pi)^2
    (a^4)/2.

    Is this a standard observation?
     
  2. jcsd
  3. Sep 23, 2008 #2

    mathwonk

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    so as far as we know, nobody has noticed this rather obvious fact for the last 2000 years?

    i am not sure actually archimedes did not know this, but it seems to escape current calc texts.
     
  4. Sep 24, 2008 #3

    mathwonk

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    in case you have not done this calculation, using slices means you have to integrate an odd power of (a^2 - x^2)^(1/2) to do even dimensional ball volumes, while using shells yields an even power.
     
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