Average value of a function over involving triple integral

[TranscendArcu]

Homework Statement

We define average value of function over a solid to be f = 1/Volume int int int f(x,y,z) dV

So find the average value of the function f(x,y,z) = x^2z+y^2z over the region enclosed by paraboloid z = 1-x^2-y^2 and the plane z = 0

The Attempt at a Solution

Actually, solving this (I don't think) is much of a problem. I get answer 1/12. But I can't intuitively understand what is going on when/if I convert to polar to solve this problem.

For example, I convert z = 1 - x² - y² and get z = 1 - r²

Why should integrating this region (multiplied by r) in the boundaries 0≤r≤1 and 0≤θ≤2*pi give the region under f(x,y,z) = x^2z+y^2z? How is the information f(x,y,z) = x^2z+y^2z incorporated into these boundaries? Or have I actually done this wrong?

 

[TranscendArcu]

Oh! Wait! We're not trying to find the volume under f(x,y,z) = x^2z+y^2z, are we? Indeed, we want to find the volume of z = 1-r^2. Right?

Okay, I think this makes intuitive sense.