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Triple Integrals (volumes)

  • Thread starter Bob Ho
  • Start date
18
0
1. Homework Statement
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
 

gabbagabbahey

Homework Helper
Gold Member
5,001
6
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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