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Thread: spherical coordinates?

  1. #1
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    spherical coordinates?

    Integrate the function
    f(x,y,z)=6*x+5*y over the solid given by the "slice" of an ice-cream cone in the first
    octant bounded by the planes x=0 and y=sqrt(43/5)*x and contained in a sphere centered at
    the origin with radius 13 and a cone opening upwards from the origin with top radius 12."
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  2. #2
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    Look at the hand-drawn diagram below.

    The diagram on the left is the shape which we are integrating. You can see the sphere of radius 13 outside. And a red cyclinder of radius 12 inside. The blue line is the line drawn from the origin to the place where the cylinder and sphere meet. The angle $\displaystyle \phi$ is the angle made by the z-axis and the blue line. Using some trigonometry (on the triangle formed by the black,red, and blue lines) we find that $\displaystyle \phi = \tan^{-1} (12/5)$.

    The diagram to the right is the picture of the region over which we are integrating. The red line is the graph of the line and the angle $\displaystyle \theta$ is the angle between the x-axis and the red line (graph). Which is $\displaystyle \theta = \tan^{-1} (\sqrt{43/5})$.

    In spherical coordinate we know that $\displaystyle x = \rho \sin \phi \cos \theta$ and $\displaystyle y = \rho \sin \phi \sin \theta$.

    Thus,
    $\displaystyle \iint_V 6x+5y dV = \int_0^{\theta} \int_0^{\phi} \int_0^{13} (6\rho \sin \phi \cos \theta + 5\rho \sin \phi \sin \theta) (\rho \sin^2 \phi) \ d\rho \ d\phi \ d\theta$.
    Attached Thumbnails Attached Thumbnails spherical coordinates?-picture.gif  
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  3. #3
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    You don't know how much you've helped, thank you.
    Sorry if I irritated you or anyone else, typical newbie move I guess
    Ellen
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