Center of mass and triple integrals

[Daniel_Feldman]

1) Find the center of mass of the tetrahedron bounded by the planes y=0, x=0, z=0, and x+3y+2z=6. p(x,y,z)=z

So I set x and y equal to 0 and got z=3, so those were my outer limits. Then I set x=0 and got y=2-(2/3)z, so that was my upper limit for y. Then x=6-2z-3y, so that was my inside upper limit. So the triple integral for the mass of the tetrahedron was int(0 to 3) int(0 to 2-2z/3) int(0 to 6-3y-2z) z dxdydz, I integrated that and got 3/2. I now need to find Mxy, Myz, and Mxz, but this would also result in three more messy triple integrals. Is there any faster way to do this?

2)Sketch the region of integration for the integral

int(0 to 3) int(9-x^2 to 9) int(0 to 9-y) f(x,y,z)dzdydx

Rewrite this as an equivalent iterated integral in three of te five possible other orders.

I mostly need help sketching the solid. I have an idea as to how it looks in the xy plane, but not in xz or yz.

 

[Daniel_Feldman]

Also, is there any sort of symmetry I can use on the first problem to make things easier?