Find Volume b/w 2 Surfaces: x, y, z Equations

[Kuma]

Homework Statement

Find the volume between the two surfaces

z = 2x^2 + y^2

z = 4 - y^2

Homework Equations

The Attempt at a Solution

Ok so i found out that the surfaces intersect at a circle. When i solved i got

x^2 + y^2 = 2, so the circle has a radius of sqrt 2.

So these are the bounds i got. X goes from - sqrt 2 to sqrt 2, y goes from - sqrt 2-x^2 to sqrt 2-x^2 and z goes from 4-y^2 to 2x^2 + y^2.

Is that right? If so, what is the equation I am integrating? Is it always integrating 1 dx dy dz?

 

[HallsofIvy]

That would certainly work. And what would be the bounds on the z-integral?


(Also, because of the circular symmetry, I would be inclined to use cylindrical coordinates- but you certainly can do it the way you suggest.)