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Volume in spherical coordinates

  1. May 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Calculate volume of the solid region bounded by z = √(x^2 + Y^2) and the planes z = 1 and z =2


    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 2, 2012 #2
    Edit: You could visualize it and integrate over 1 and add these volumes.
     
    Last edited: May 2, 2012
  4. May 2, 2012 #3
    it's a cone, but how do you set the limits for the different integrals in spherical coordinates?
     
  5. May 2, 2012 #4
    In sphereical coordinates you know that [itex]x=\rho\cos\theta\sin\phi[/itex], [itex]y=\rho\sin\theta\sin\phi[/itex] and [itex]z=\rho\cos\phi[/itex]
    You can use this to find limits for [itex]\rho[/itex].
    If you draw the x-z or y-z plane intercept this can help you find [itex]\phi[/itex]
     
  6. May 2, 2012 #5

    sharks

    User Avatar
    Gold Member

    You should first plot it to know what the volume looks like.

    The volume between z=1 and z=2 is that of a circular disk. You need to use cylindrical coordinates.

    Description of the region:
    For r and θ fixed, z varies from z=1 to z=2
    For θ fixed, r varies from r=1 to r=√2
    θ varies from θ=0 to θ=2∏

    Plug the limits into the triple integral and evaluate to find the required volume:
    [tex]\int \int \int dr d\theta dz[/tex]
     

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