1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

    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

    tiny-tim

    User Avatar
    Science Advisor
    Homework Helper

  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

    SteamKing

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper

  9. Jul 25, 2013 #8

    vanhees71

    User Avatar
    Science Advisor
    2016 Award

    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
    [tex]x=\frac{1}{\sqrt{y^2+z^2}}=\frac{1}{r}.[/tex]
    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

    http://en.wikipedia.org/wiki/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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Triple integral problem: cylindrical coordinates
Loading...