If it were me, I'd want a rough estimate before doing too much. This might suggest whether one had wandered off.
The intersection of the two surfaces suggests as a lower bound (don't forget the bottom of your container) and not much more than that as an upper bound. Certainly less than
A quick check of your Rectangular Coordinates results in about and your last expression comes in at a whopping .
Seems like we don't quite have the right things going on just yet.
On the other hand, maybe I'm coming off the wrong medication and making far less sense than I think.