Confused about polar integrals and setting up bounds

[mncyapntsi]

Homework Statement

I am trying to find the volume between y = x^2+z^2 and y = 3-4x^2-2y^2.

Relevant Equations

x^2+y^2+z^2 = p^2
z^2+y^2 = r^2

So I want to subtract the two surfaces, right? I really don't know where to start... I am guessing this would be some sort of triple integral, however I am very confused with the bounds.
Any help would be greatly appreciated!
Thanks!!

 

[WWGD]

I assume the 3rd line, where z=8-x^2-x^2 has a mistake? Also, please write your equations in Latex.

 

[Mark44]

I assume the 3rd line, where z=8-x^2-x^2 has a mistake?

I don't see a 3rd line or this equation. Likely the OP deleted it.

 

[WWGD]

I don't see a 3rd line or this equation. Likely the OP deleted it.

I think the problem statement was edited.

 

[Frabjous]

The first surface is a sphere. The second surface is a cylinder. So I am guessing that p²>r² and the question is actually what volume of the sphere is outside the volume of the cylinder.

Is this correct?
Given this geometry, which type of coordinate system (cartesian, cylindrical or spherical) is appropriate for integration and why? Hint: think about cross sections. You might want to draw a figure.