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Triple Integral Using Cylindrical Coordinates

  1. Feb 24, 2010 #1
    1. The problem statement, all variables and given/known data
    A conical container with radius 1, height 2 and with its base centred on the ground
    at the origin contains food. The density of the food at any given point is given by
    D(r) = a/(z + 1) where a is a constant and z is the height above the base.
    Using cylindrical polar coordinates, calculate the total mass of food in the container.


    2. Relevant equations



    3. The attempt at a solution
    ok so mass is the integral D(r)dV, and in cylindrical coordinates dV is rdrd[tex]\theta[/tex]dz

    I thought that you could probably do:
    [tex]\int^1_0 \,dr[/tex][tex]\int^{2\pi}_0 \,d\theta[/tex][tex]\int^{2r-2}_0 \,dz[/tex]

    [tex](ra/(z+1))[/tex]

    But this makes the integral very difficult and I don't think it's right. I'm pretty sure there's something wrong with my limits on dz. Any help would be appreciated
     
  2. jcsd
  3. Feb 24, 2010 #2

    tiny-tim

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    Hi henryc09! :smile:
    Yes, you've integrated from z = 0 to z = 2r - 2 …

    why?? :confused:

    z doesn't depend on r, the limits of z are the same for all r. :wink:
     
  4. Feb 24, 2010 #3
    OK so how do I work out the limits of z? It can't just be from 0-2 because that would make it a cylinder? Still a bit confused.
     
  5. Feb 25, 2010 #4

    tiny-tim

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    Hi henryc09! :smile:

    (just got up :zzz: …)
    I'm sorry … somehow I read it as a cylinder. :redface:

    Yes, your limits were correct.

    They should lead you to an integral of rlog(2r-1) … you can use integration by parts on that.
     
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