Did you try to make some graph?
Note z=x^2+y^2 is a parabolid which opens up and the vertex is on the origin. And 3z=4-x^2-y^2 is a parabolid which opens down and vertex is at (0,0,4/3) if you divide by 3 to bring it to the form z=f(x,y). Thus, the second one is the top one.
How do they intersect? It looks like, if you graph them, that they intersect in a circle. We can actually show that,
Then substitute equation (1) into equation (2):
3z = 4 - z
Thus, they intersect at z=1 (in that plane) and if you substitute that value in any equation say (1) then you get:
You get a circle with radius one.
In order to find the volume between these paraboloids we also need to know their projection onto the xy plane. Which is what we found above, it is the circle with radius 1.
Now we can find the volume.
This is Mine 54th Post!!!