How Do You Calculate the Average Temperature of a Solid Using Triple Integrals?

[Bob Ho]

Homework Statement

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.

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]

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 $T(z)$