Calculating the Volume of a Concave Lens Using Multiple Integration

[gboff21]

Homework Statement

A circular concave lens of radius 2 units, has two refracting surfaces described by z=1/2(x2+y2+1) and z=-1/2(x2+y2) What is the volume of the glass?

Homework Equations

Over a region R: V=$$\int\int\int dA$$

The Attempt at a Solution

I have no idea how to start this. Polar coordinates, bounded regions I don't know anything apart from the fact that I need to use multiple integration.

 

[HallsofIvy]

A line from $z= (1/2)(x^2+ y^2)$ straight up (parallel to the z axis) to $z= (1/2)(x^2+ y^2+ 1)$ has length 1 so the volume is simply the integral of "1" over the region, in the xy plane, the lens covers. And that is just the 1 times the area of that region! What is the area of a circle of radius 2?

 

[gboff21]

The second surface is $z= -(1/2)(x^2+ y^2+ 1)$. Its negative. Even so, I'm doing this question for an exam on monday out of practise for multiple integrals. So how would you do this using integration?