Finding the Volume of a tetrahedron using Spherical Coordinates

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The discussion revolves around calculating the volume of a tetrahedron defined by the plane equation 3x + 2y + z = 6 in the first octant using spherical coordinates. The volume is confirmed to be six. Participants are attempting to determine the correct boundaries for the spherical coordinates, with p ranging from 0 to a specific function of phi and theta. There is some uncertainty about the limits for theta, which is believed to be between 0 and pi/2, while phi is clarified to range from 0 to pi/2. The conversation highlights the challenges in establishing these boundaries for the volume calculation.
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Find the volume of a tetrahedron under a plane with equation 3x + 2y + z = 6 and in the first octant. Use spherical coordinates only. The answer is six.




x=psin(phi)cos(theta)
y=psin(phi)sin(theta)
z=pcos(phi)



I've been trying to figure out the boundaries of this particular problem all night. To be honest, I'm completely at a loss. I have p going between 0 and (6/(cos(phi)+sin(phi)(3cos(theta)-2sin(theta)). I believe that theta is between 0 and pi/2, although I'm not entierly sure on that one. As far as phi goes I believe the upper limit is pi but the lower limit is a mystery to me.
 
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\phi would go from 0 to \pi/2 as does \theta.
 

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