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Homework Help: Triple integral problem: cylindrical coordinates

  1. Jul 25, 2013 #1
    1. The problem statement, all variables and given/known data
    I have a graph 1/x^2=y^2+z^2 where z=rsin(θ) and y=rcos(θ)
    where 0≤r≤1 and 0≤θ≤2∏ on the zy-plane

    The end result is attached (sorry, I'm not aware of how to use Latex :[ )
    I can kind of understand how they determined the first bounds for the integral: the lowest x co-ordinate is 0 and the maximum co-ordinate is always 1/r. I can also kind of understand how they determined the second integral bounds, the maximum possible value for r is 1 and the lowest possible value for r is 0. For the angle the largest possible angle is 2∏ whereas the lowest possible angle is 0. I do not, however, understand why there is a half in front of the triple integral.

    Attached Files:

  2. jcsd
  3. Jul 25, 2013 #2


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    hi jackscholar! :smile:

    i don't understand what the question is …

    is it to find a volume, if so of what?
    surely it's from 1 to 1/r ? :confused:
  4. Jul 25, 2013 #3
    I am trying to determine the volume for Torricelli's Trumpet. I've done it the conventional way but have been want to prove it using the triple integral method. I stumbled across a website which stated a few things and got to the answer but it didn't explain how. I have proved the surface area using a different method to the disk method and would like to prove the volume using a different method but I'm not too good at triple integrals. The website is below, what I was looking at was on page 18 near the bottom and 19 near the top. Could you help m interpret what they are trying to do?

    This is the website: http://www.palmbeachstate.edu/honors/documents/jeansergejoseph.pdf
  5. Jul 25, 2013 #4


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  6. Jul 25, 2013 #5
    It's easy: it's just the volume of ONE of the horns.
  7. Jul 25, 2013 #6
    How would I calculate the volume of one of the horns using a triple integral, though?
  8. Jul 25, 2013 #7


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  9. Jul 25, 2013 #8


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    You use the cylindrical coordinates as given in #1:
    [tex]\vec{r}=(x,r \cos \varphi,r \sin \varphi).[/tex]
    Then the boundary surface for one half of the horn is given as
    Now the volume element in cylinder coordinates is
    [tex]\mathrm{d}^3 \vec{r}=\mathrm{d} r \mathrm{d} \varphi \mathrm{d} x r.[/tex]

    Now using the definition for Gabriel's horn


    It's volume is given by
    [tex]V=\int_0^1 \mathrm{d} r \int_0^{\varphi} \mathrm{d} \varphi \int_{1}^{1/r} \mathrm{d} x r.[/tex]
    Now you can easily evaluate the integral yourself.
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