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Volume Integral of Cone

  1. Feb 5, 2009 #1
    1. The problem statement, all variables and given/known data
    "A solid cone is bounded by the surface [tex]\theta=\alpha[/tex] in spherical polar coordinates and the surface [tex]z=a[/tex]. Its mass density is [tex]p_0\cos(\theta)[/tex]. By evaluating a volume integral find the mass of the cone.


    2. Relevant equations



    3. The attempt at a solution
    I can't figure out the correct limits for the volume integral. Is it best to solve in Cartesian or spherical polar coordinates?

    Many thanks :)
     
  2. jcsd
  3. Feb 5, 2009 #2
    Your description of the cone suggests your interpretation of spherical polar coordinates is [tex](r, \theta, \phi)[/tex] where [tex]\theta[/tex] is the angle from the positive z-axis and [tex]\phi[/tex] is the angle from the positive x-axis.

    We look to use these coordinates to calculate the integral for the cone. Sketch the cone: it makes an angle of alpha with the positive z-axis and goes up to z=a. More specifically...

    [tex]\theta[/tex] runs from [tex]0[/tex] to [tex]\alpha[/tex].

    [tex]\phi[/tex] goes from ... to ... ?

    To find the r-limits, draw a right-angled triangle:
    Code (Text):
       -----
       |   /
     a |  / r      where A is the angle alpha.
       |A/
       |/
    .
     
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