Volume between 2 paraboloids

Today was my test and I've absolutely no idea about if I'll pass it or not. (On the cut, or I don't know how to say)

I had to calculate the volume between the paraboloids $\displaystyle z=x^2+y^2$ and $\displaystyle z=\frac{4}{3}-\frac{x^2}{3}-\frac{y^2}{3}$, if I remember well.

I've thought about it and I think that using Cartesian coordinates will be easier than spherical and cylindrical ones.

However I tried to calculate the volume via cylindrical coordinates: $\displaystyle (r,\theta,z)$. I tried to set up the triple integral : $\displaystyle z$ goes from $\displaystyle r^2$ to $\displaystyle \frac{4}{3}-\frac{r^2}{3}$.

$\displaystyle \theta$ from $\displaystyle 0$ to $\displaystyle 2 \pi$.

And $\displaystyle r$ goes from $\displaystyle 0$ to .... I wasn't sure. But I strongly believe I already made an error.

I have 2 questions : How can I calculate the volume they ask using cylindrical coordinates?

With what method would you solve the problem?

Thank you very much!

Oh, and before I forget, I know they cross each other with a circular ($\displaystyle x^2+y^2=1$) section.