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Volume integral of cone

  1. Oct 11, 2011 #1
    I have the cone x^2 + y^2 <= z^2 with |z| <= 2
    The vector function F = (4x, 3z, 5y)

    With the divergence theorem I managed to reduce the equation to
    ∫∫∫ 4 dxdydz

    Now the problem is finding out the limits. I know z goes from 0 to 2, but what about x and y?
     
  2. jcsd
  3. Oct 11, 2011 #2

    LCKurtz

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    You haven't stated the problem. Are you calculating a flux integral? You might try writing your volume integral in cylindrical coordinates.
     
  4. Oct 11, 2011 #3
    The question asks to evaluate ∫F.ndA by the divergence theorem. I can just take a shortcut and use the general formula for cone volume, but I was wondering if there are known limits for the integrals.
     
  5. Oct 11, 2011 #4

    LCKurtz

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    Of course there are. Integrate z first from z on the cone to z on the top and use polar coordinates for the dxdy integral. That is why I suggested cylindrical coordinates.
     
  6. Oct 12, 2011 #5

    SteamKing

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    Your integral is very simple and can be evaluated without explicitly determining the limits of each integral. (Hint: what is the integral of dV of a cone over the entire volume of the cone?)
     
  7. Oct 12, 2011 #6

    LCKurtz

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    Apparently he already knows that.
     
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