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Triple Integral - Volume of Tetrahedron

  1. Apr 30, 2012 #1
    1. The problem statement, all variables and given/known data

    Actually, the problem was addressed in a prior post:

    https://www.physicsforums.com/showthread.php?t=178250

    Which is closed.


    2. Relevant equations


    I would like to know how HallsofIvy (or anyone) arrived at the formula for the tetrahedron given the vertices (1,0,0), (0,2,0), (0,0,3).

    Ultimately I am to find the volume of this tetrahedron using triple integrals.

    But I'm not worried about the integral as much as the setup:

    The equation I get is 3 -3x -3/2y
    not 1 -3x -3/2y


    3. The attempt at a solution
    I've taken two vectors from these points, taken their cross product, and created an equation of a plane. I am still getting my answer, and consequently an integral that doesn't seem right!


    -Dave K
     
  2. jcsd
  3. Apr 30, 2012 #2

    sharks

    User Avatar
    Gold Member

    You should label each of the vertices. Let A=(1,0,0), B=(0,2,0), C=(0,0,3). There are 2 methods (that i know of):

    First method:
    The Cartesian equation of plane is: [itex]ax +by+cz=d[/itex]

    Just plug in the coordinates of the 3 points and solve the system of 3 linear equations. You should get x, y and z in terms of d. Then, divide throughout by d to get the final equation of the plane.

    Second method (what you're expected to use):

    Find two vectors that lie in the plane. Do the cross product to get the normal vector, [itex]\vec n[/itex].

    Then, use the formula: [itex]\vec r.\vec n=\vec a.\vec n[/itex] where [itex]\vec a[/itex] is any point found in the plane.

    Using the second method, you will get the Cartesian equation of the plane: [itex]6x+3y+2z=6[/itex]

    There is indeed a mistake in that post: https://www.physicsforums.com/showpost.php?p=1387411&postcount=3

    [tex]z=3-3x-\frac{3y}{2}[/tex]
     
    Last edited: Apr 30, 2012
  4. Apr 30, 2012 #3
    Oh thank goodness.

    I wasn't trusting my own answer, and it (the correct answer) makes the integration a little bit uglier.

    Thanks man.

    Regards,

    -Dave K
     
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