I'm confident with solving this question in cylindrical coordinates (for which there are few answers) but I'm doubtful about cartesian coordinates and can't seem to figure out the limits. $z$ needs to be the outer integral in the below question:

[![enter image description here](https://i.stack.imgur.com/X1FbI.png)](https://i.stack.imgur.com/X1FbI.png)
[![enter image description here](https://i.stack.imgur.com/OaN7M.png)](https://i.stack.imgur.com/OaN7M.png)

Right now, I've figured that:

1. $z$ must vary from $-1$ to $1$ (since it describes the radius of the cylinder).
2. If we make $x$ the inner integral and dependent on $y$ we could say that it varies from $-2y^2$ to $(1-2y^2)$ (obtained by substituting the cylinder equation and adding in $a=0$ & $a=1$
3. I'm not sure how to evaluate the middle integral and how exactly there are 2 regions over which we must integrate.