Are you trying to find the centre of mass or are you attempting to find the expectation (mean) of a distribution over three dimensional space?
The question: "Find the average z-value of points in the ice cream cone shaped region common to the sphere x2 + y2 + z2 = 4 and the cone z = sqrt(3(x2 + y2)). I know how to find the "average value" of a function, but I don't quite understand what my professor wants when she says the "average z-value".
For those who might need to know for some reason, I did find the volume of "ice cream cone shaped region" which she has mentioned:
Note: I used spherical co-ordinates (and I'm fairly sure I'm right)
This problem isn't geared towards any physical application. I have presented the question as I received it, literally. If it seems nonsensical, it isn't the first time there is a typo in the assignment.
You evaluated the integral correctly, but is calculating the volume of the "ice cream cone". To get the average value of z, you need to calculate the same integral with z and divide. So since :