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Math Help - Triple integral (How to set up)

  1. #1
    VkL
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    Triple integral (How to set up)

    Use a triple integral in rectangular coordinates to find the mass of the region bounded by the planes x+y+z=1, x=0, y=0, and z=0 in octant 1, if the density at each point (x,y,z) is p(x,y,z)=kx

    How do you set this up??
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  2. #2
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    Always try to draw it first. If x+y+z=1 then when y=0, you have the line z=1-x in the z-x plane and if x=0, you have z=1-y in the z-x plane. Now just connect the lines in the x-y plane between y=1 and x=1. Try and draw those three lines in an x-y-z coordinate axis on paper then try to understand why the integral is (I believe):

    \int_0^1\int_0^{-x+1}\int_0^{1-y-x} kx dzdydx
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  3. #3
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by VkL View Post
    Use a triple integral in rectangular coordinates to find the mass of the region bounded by the planes x+y+z=1, x=0, y=0, and z=0 in octant 1, if the density at each point (x,y,z) is p(x,y,z)=kx

    How do you set this up??
    It's been a while since I've done these, but I think it's:

    The solid shape we are integrating over is a tetrahedron; having a geometric sense can help with determining the limits of the definite integrals. For general region R, the triple integral is

    \iiint\limits_R f(x,y,z)\, dV\ = \int\limits_{a}^{b}\int\limits_{c}^{d}\int\limits_  {e}^{f}f(x,y,z)\,dx\,dy\,dz

    where we can integrate in any order. Here we will want to integrate with respect to x first, because p(x,y,z) only depends on x.

    Anyway I get:

    \int\limits_{0}^{1}\int\limits_{0}^{(1-z)}\int\limits_{0}^{(1-y-z)}(kx)\,dx\,dy\,dz

    I'm pretty sure I got the upper limits right but not 100%, anyway you might find this site helpful, example 4 deals with a more generalized tetrahedral region defined similarly.

    EDIT: maybe shawsend is right and it's easier to integrate dx last... it's been too long... sorry

    EDIT 2: integrating with respect to x last is definitely easier.
    Last edited by undefined; April 5th 2010 at 03:17 AM.
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