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Math Help - Cylindrical Coordinates Triple Integral

  1. #1
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    Cylindrical Coordinates Triple Integral

    Hey everyone,

    I am given the integral:

    Evaluate the integral of f(x,y,z) = z(x^2 +y^2 +z^2)^(-3/2) over the part of the ball x^2 +y^2 +z^2 less than or equal to 81 defined by z greater than or equal to 4.5.

    This is the top of the sphere. I know that theta runs from zero to 2pi, but I am very confused about how to find the angle phi, which runs from the z axis to a point P, and the distance from the origin to the point P, which we call Ro.

    I need to find what they run between to calculate my integral. Any help would be appreciated. Thanks.
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  2. #2
    MHF Contributor

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    z= 4.5 intersects the sphere x^2+ y^2+ z^2= 81 where x^2+ y^2+ (4.5)^2= 81 or x^2+ y^2= 81- 20.25= 60.75, a circle with center at (0, 0, 4.5), radius \sqrt{60.75}= 7.79. That is, the straight line from the origin to a point on that circle is the hypotenuse of a right triangle with "opposite side" 4.5 and "near side" 7.79. phi is the arctan of \frac{4.5}{7.79}.
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