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Math Help - Mass of a sphere, checking my result

  1. #1
    MHF Contributor arbolis's Avatar
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    Mass of a sphere, checking my result

    Let B be a ball with radius a and in which the density of any point is equal to the distance to a fixed diameter. Find the mass of the ball.

    My attempt : m=\iiint _{B} \rho (r,\phi, \theta) drd\phi d \theta = \int_0^{2\pi} \int_0^{\pi} \int _0^a r^3 \sin (\phi) drd\phi d\theta=a\pi.
    I had a hard time finding that \rho (x,y,z)=x^2+y^2 if I chose the fixed diameter as the z-axis. Then I simply converted into spherical coordinates chosing " r" instead of " \rho" because I already got confused in another exercise because of the notation.
    Is my result correct?
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  2. #2
    Super Member Random Variable's Avatar
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    Do they mean the shortest distance?

    If that's what they mean, the shortest distance from a point to the z-axis is  \sqrt{x^{2}+y^{2}} = r \sin \phi .
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  3. #3
    MHF Contributor Calculus26's Avatar
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    See attachment--I think you are also confusing the cylindrical coordinate r.
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  4. #4
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Random Variable View Post
    Do they mean the shortest distance?

    If that's what they mean, the shortest distance from a point to the z-axis is  \sqrt{x^{2}+y^{2}} = r \sin \phi .
    I guess yes.
    Ah yes, I knew it was \sqrt{x^2+y^2} but I don't know why I wrote x^2+y^2, even in my draft!

    Quote Originally Posted by Calculus26 View Post
    See attachment--I think you are also confusing the cylindrical coordinate r.
    Oops, right. Ahmm... poor me.
    Calculus 26, \delta (x,y,z)=\sqrt{x^2+y^2} as Random Variable pointed out.
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  5. #5
    MHF Contributor Calculus26's Avatar
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    thanks that makes the integrand p^3 sin^2(phi) etc...
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