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Math Help - triple integrations

  1. #1
    Econ_time
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    triple integrations

    1. consider the solid tetrahedron determined by (0,0,0), (0,0,1),
    (0,root13,0), (root2, root13, 0). SET UP the triple integrations so that its volume = triple integral of dy*dx*dz, and triple integral of dz*dy*dx.
    is asking for the limit boundaries.
    please explain step by step, i have the answers, but i dont' know how to get there.

    2. find the center of mass of the solid above z = root(x^2+y^2) and below x^2 + y^2 + z^2 =4, and the density at a point equals the z-coordinate of the point.
    i believe the center of mass = moment/mass.
    with this problem please also explain step by step, and state all the equations you used.

    THX!!!
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  2. #2
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    Quote Originally Posted by Econ_time View Post
    1. consider the solid tetrahedron determined by (0,0,0), (0,0,1),
    (0,root13,0), (root2, root13, 0). SET UP the triple integrations so that its volume = triple integral of dy*dx*dz, and triple integral of dz*dy*dx.
    is asking for the limit boundaries.
    please explain step by step, i have the answers, but i dont' know how to get there.
    Look at hand drawn picture below.

    As you know the volume is,
    \int\int_S\int dV
    Where, S is the set of points determined by this tetrahedron.
    By Fubini's theorem it can be computed as,
    \int_A \int \int_{u(x,y)}^{v(x,y)} dz\, dA
    Where,
    u(x,y) is the lower surface bound on the tetrahedron in this case u(x,y)=0 and v(x,y) is the equation of the plane above. To find the equation of that plane we need to find the equation of the plane passing through the points:
    (0,0,1), (0,\sqrt{13},0), and (\sqrt{3},\sqrt{13},0)
    I am not going to find the equation of the plane that is for thee to find.

    Next, you need to express the region as Type I and Type II:
    In that case,
    \int_0^{\sqrt{3}} \int_{\sqrt{13/3}x}^{\sqrt{13}} \int_0^{v(x,y)} dz\, dy\, dx
    And,
    \int_0^{\sqrt{13}} \int_0^{\sqrt{3/13}y} \int_0^{v(x,y)} dz\, dx\, dy

    Again where,
    v(x,y)
    Is the equation of plane expressed in terms of x,y
    Attached Thumbnails Attached Thumbnails triple integrations-picture12.gif  
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