Finding the Volume of a tetrahedron using Spherical Coordinates

[Kincaid]

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.

 

[LCKurtz]

$\phi$ would go from 0 to $\pi/2$ as does $\theta$.