Volume of a solid bound by four surfaces

[Conservation]

Homework Statement

Compute the volume of the solid bounded by the four surfaces x+z=1,x+z=−1,z=1−y²,z=y²−1

Homework Equations

Fubini's theorem?

The Attempt at a Solution

I have tried to visualize this solid and define the limits; when I attempted to integrate by dxdzdy (in that order), I set the limits for the first integral as -z-1 to -z+1, the second integral as y^2-1 to 1-y^2, and third integral as -1 to 1, but the I'm not getting a value due to the second integral evaluating as zero.

I feel almost certain that the problem mainly has to do with defining the correct limits and order of integration, but I'm having a bit of trouble here.

Thanks.

 

[mfb]

How can the second integral be zero? Some sign error?
The order of integrals is useful.

 

[Conservation]

Because y²-1=-(1-y^2) and there are no y terms after the first integral, the integral equals 0.

 

[mfb]

If you integrate f(y)=1 from y=-5 to y=5, is the result 0 - or maybe 10?

 

[Conservation]

Wow. I can't believe that I made such a dumb mistake. I stared at it for at least 10 min straight.

Thanks for putting up with such a trivial problem. X(