I have been attempting this question for more than four hours now. Since I don't have the answer key for it, I try and check my answer using Symbolab by ensuring all three integrals give the same volume. However, I haven't managed to get the that to happen till now.
For part (a): $z$ should range from $x^2 \to 3-y$, $x$ should range from $-\sqrt{3-y} \to \sqrt{3-y}$, and $y$ should range from $0 \to 2$. Are these correct for this part?
Following the confirmation that my answer for part (a) is right: For part (b): We can split it into two integrals:
$x$ should range from $-\sqrt{z} \to \sqrt{z}$, $y$ should range from $0 \to 3-z$ and $z$ should range from $1 \to 3$.
$x$ should range from $-\sqrt{z} \to \sqrt{z}$, $y$ should range from $0 \to 2$ and $z$ should range from $0 to 1$.
Is this also correct?