Help finding boundries for integration

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I'm trying to find the volume of the solid bounded by $y=x^3$, $y=2-x$ I have $0<=y<=1$ and $y^(1/3)<=x<=2-y$. Is there something wrong with this? I was also trying to find the volume of the solid bounded by $x^2 + y^2 + x^2 = 2$ and $z=x^2+y^2$. I know what z is bounded by, but i have no idea for y or x.