# Triple Integrals (volumes)

1. Jun 14, 2009

### Bob Ho

1. The problem statement, all variables and given/known data
A solid is definited by the inequalities 0$$\leq$$x$$\leq$$1, 0$$\leq$$y$$\leq$$1, and 0$$\leq$$z$$\leq$$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. Jun 14, 2009

### gabbagabbahey

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

$$\langle f \rangle \equiv \frac{\int_{\mathcal{V}}f dV}{\int_{\mathcal{V}} dV}$$

...apply that to $T(z)$