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Math Help - The average z-value of a points in a R3 region

  1. #1
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    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)

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


    -Giest
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  2. #2
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    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?
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  3. #3
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    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.
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  4. #4
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    Re: The average z-value of a points in a R3 region

    Is this from a statistics/probability class?
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  5. #5
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    Re: The average z-value of a points in a R3 region

    No, it is from a vector calculus class.
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  6. #6
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    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
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