It seems like you're on the right track, but since it says OUTSIDE of the cylinder, I believe you want the region instead.
I cannot seem to make heads or tails of this problem.
A volcano fills the volume between the graphs and , and outside the cylinder . Find the volume of this volcano.
I converted it into polar form which resulted in Integral[Integral[(1/r^16)r drd(Theta)]] bounded by r=0, r=1, Theta=0, Theta=2Pi