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Integral in cylindrical coordinates

  1. Apr 9, 2012 #1
    1. The problem statement, all variables and given/known data

    I need to calculate the integral where the region is given by the inside of x^2 + y^2 + z^2 = 2 and outside of 4x^2 + 4y^2 - z^2 = 3

    2. Relevant equations



    3. The attempt at a solution

    So far, I think that in cylindrical coordinates (dzdrdtheta):

    0 <= theta <= 2pi
    sqrt(3)/2 <= r <= 1
    -sqrt(2-r^2) <= z <= sqrt(2-r^2)

    Are the bounds for the radius and z correct?
     
  2. jcsd
  3. Apr 9, 2012 #2

    LCKurtz

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    Do you know what this figure looks like? If you look at the surfaces you should see that you don't want to do the dz integral first, on the inside. Do you see why? And don't forget the ##r## in your cylindrical volume element.
     
  4. Apr 9, 2012 #3
    In the order drdzdtheta, i get:

    0 <= theta <= 2pi
    sqrt(z^2 + 3)/2 <= r <= sqrt(2-z^2)
    -1 <= z <= 1

    I understand your point, the radius varies from z = -1 to z = 1 because of the hyperboloid, but the exercise is asking me to give the integrals for both orders: dzdrdtheta, drdzdtheta

    Thanks for the help.
     
  5. Apr 9, 2012 #4

    LCKurtz

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    If you look at a cross-section in the ##z-r## plane you will see that ##z## is a two-piece function of ##r## on both the top and bottom. So doing ##z## first will require two integrals for ##dz##.
     
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