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Find volume within sphere outside of Cylinder

  1. Aug 2, 2009 #1
    1. The problem statement, all variables and given/known data
    Find the volume of the solid that lies within the sphere x^2 + y^2 + z^2 = 16, above the xy plane, and outside the cone z = 3 [tex]\sqrt{{x^2+y^2}}[/tex].


    2. Relevant equations
    spherical system:
    x=[tex]\rho[/tex]cos[tex]\theta[/tex]sin[tex]\phi[/tex]
    y=[tex]\rho[/tex]sin[tex]\theta[/tex]sin[tex]\phi[/tex]
    z=[tex]\rho[/tex]cos[tex]\phi[/tex]

    cylindrical system
    [tex]x^2+y^2=r^2[/tex]
    z=z

    3. The attempt at a solution
    I have tried this using both the spherical and cylindrical systems and arrived at the same answer, cylindrical is easiest here, so Ill use it to demonstrate what I have done

    [tex]x^2+y^2+z^2=16[/tex]
    [tex]r^2+z^2=16[/tex]
    z=[tex]\sqrt{{16-r^2}}[/tex]

    z = 3 [tex]\sqrt{x^2+y^2}[/tex]
    z=3[tex]\sqrt{r^2}[/tex]
    z=3r

    when z=0
    0=[tex]\sqrt{16+r^2}[/tex]
    r=4
    0=3r
    r=0

    Therefore the bounds are
    z: [tex]\left[3r,\sqrt{16-r^2}\right][/tex]
    r:[tex]\left[0,4\right][/tex]
    [tex]\theta[/tex]: [tex]\left[0,2\pi\right][/tex]

    Which gives me
    [tex]\int_{0}^{2\pi}\int_{0}^{4}\int_{3r}^{\sqrt{16-r^2}}r dzdrd\theta[/tex]

    After going through the steps i get
    [tex]-(256\pi)/3[/tex]

    First off it is negative, so that tells me I am completely off base here, but when doing the same problem with spherical coordinates, I get [tex](256\pi)/3[/tex]. Since this answer is wrong, I am at a loss of what I should be doing. Please help.
     
    Last edited: Aug 2, 2009
  2. jcsd
  3. Aug 2, 2009 #2
    I solved it:

    First the bounds had to be reworked for r in terms of Z
    r: [tex]\left[z/3,\sqrt{16-z^2}\right][/tex]

    Then by solving the two equations in terms of Z and setting them equal to each other, I determined the maximum value for r
    [tex]\sqrt{16-r^2}=3r[/tex]
    [tex]r=(2*\sqrt{10})/5[/tex]
    after plugging that in and solving for z, z's maximum is 3.83863112788

    Then using that as the bounds for Z and using the same bounds for [tex]\theta[/tex]
    The answer came out to be 127.137105725

    Thanks to anyone that attempted to figure it out for me...
     
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