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Homework Help: Triple Integrals (volumes)

  1. Jun 14, 2009 #1
    1. The problem statement, all variables and given/known data
    A solid is definited by the inequalities 0[tex]\leq[/tex]x[tex]\leq[/tex]1, 0[tex]\leq[/tex]y[tex]\leq[/tex]1, and 0[tex]\leq[/tex]z[tex]\leq[/tex]x2+y2. The temperature of the solid is given by the function T=25-3z. Find the average temperature of the solid.

    3. The attempt at a solution

    I solved the integral, however I could not figure out how to determine what to do to find the average temperature value. In the answers i was given. They have no explanation, just the volume of solid above the inequalities is (!) 2/3.
    So they therefore times the integral by 3/2.

    Can someone please explain how this idea works? Thanks
  2. jcsd
  3. Jun 14, 2009 #2


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    Homework Helper
    Gold Member

    The average value of any function [itex]f(x,y,z)[/itex] over some volume [itex]\mathcal{V}[/itex] is, by definition;

    [tex]\langle f \rangle \equiv \frac{\int_{\mathcal{V}}f dV}{\int_{\mathcal{V}} dV}[/tex]

    ...apply that to [itex]T(z)[/itex]
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