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 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)