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Math Help - Evaluating integrals of a tetrahedron

  1. #1
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    Evaluating integrals of a tetrahedron

    Evaluate the triple integral where is the solid tetrahedon with vertices .

    My question is, how do you find the equation of the plane z = ?
    I need that to start solving the problem but i have no idea how to do that.
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  3. #3
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    This is what it looks like to me: You need the equation of the gray plane in the figure below. To get it, you first calculate a normal which I've shown. Calculate the normal via the cross-product of two vectors which you can calculate emanating from the point (0,4,0). Once you get the normal, then the equation of the plane is n_x(x-p_x)+n_y(y-p_y)+n_z(z-p_z)=0

    I get:

    I=\int_0^6\int_0^{f(x)}\int_0^{g(x,y)} xydzdydx

    with f(x)=-2/3x+4
    g(x,y)=1/24\left(-24x-36y-144\right)

    I'm not sure about this. Hopefully, this can get you in the ball-park.
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