Using integral to find volume problem

Hi there,

I have an exercise I couldn't solve, have the answer which i couldn't get to, any guidance will be appreciated:

Calculate volume of D where D is the region in $\displaystyle R^3 $enclosed by the surfaces

z=3-x^2-y^2 and z=x^2+y^2 -3 (Sorry Latex was giving me errors)

So this looks like 2 identical parabaloids one startign at z=3 going down and one starting at z=-3 going up.

I used a triple integral with cylindrical coordinates ( if theres another better way let me know)

So I integrated over

theta = 0 to 2pi ( ie all the way around)

r= sqrt(3) by equating the 2 parabaoids to find the intesection

z integrated over 3-r^2 and r^2-3.

The first issue here is that when i do the z integration i come out with zero (what did I do wrong here)

So i decided to integrate in 2 identical parts split at z=0.

I followed this through but didn't get the books answer of 9pi. i got 24root(3)/5 for one half, how does the book get to 9 pi? At which step did I go wrong?

(if this is impossible to read I'll reattempt the Latex)

Thank you