Thread: The average z-value of a points in a R3 region

1. The average z-value of a points in a R3 region

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)

-Giest

2. Re: The average z-value of a points in a R3 region

Hey Giestforlife.

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?

3. Re: The average z-value of a points in a R3 region

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.

4. Re: The average z-value of a points in a R3 region

Is this from a statistics/probability class?

5. Re: The average z-value of a points in a R3 region

No, it is from a vector calculus class.

6. Re: The average z-value of a points in a R3 region

You evaluated the integral correctly, but $\int\int\int\, \rho^2\sin{\phi}\, d\rho\,d\phi\,d\theta$ 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 $z=\rho\cos{\phi}$:

$\overbar{z}=\frac{\int\int\int\, \rho^3\sin{\phi}\cos{\phi}\,d\rho\, d\phi\,d\theta}{\int\int\int\, \rho^2\sin{\phi}\,d\rho\, d\phi\,d\theta}$

- Hollywood