Although I've only shot a few beavers(22 magnum revolver, and 357 magnum's), I have friends that do it quite a bit. I'd rather trap'em myself
. I can tell you what I know from my limited personal experience, and what I know from their exstensive experience.
First off, do NOT use a 22 LR for beavers. As kyelkhunter3006 said, beavers can get very big. It's not uncommon at all around here to trap a 60-70 lb beaver, and they get bigger than that. Even your average beavers around here will easily run 35-50 lbs. They're also extremely tough, and can soak up a good hit, and still get away.
All the fellows I know that shoot them use either a 22 magnum, or something slightly bigger like a 22 Hornet. Around here THE beaver round in a 22 magnum is the CCI TNT HP's, or the CCI Maxi-Mag +V. Most of these guys try for a head shot, but say a good chest shot with either of these loads will put'em down better than any other 22 magnum ammo. None of the 22 magnum guys I know like the heavier bullets because they just don't expand reliably in any animal. I can vouge for that as I've shot dozens of groundhogs, and a few foxes with the CCI, and Federal 50g loads, and they were a dismal failure in a revolver, and not much better from a rifle.
I've shot a couple beavers in the head with the CCI TNT's fired from my 5.5" Single Six at around 15-20 yards, and of course, both dropped on the spot with no exit. I've also shot a few more with my 357 Blackhawk, and 357 rifle using 125g Remington SP's at 1250 fps from the Blackhawk, and around 1400-1500 fps from the rifle at ranges from 20 yards to maybe 50 yards. they were all chest/shoulder shots best I can remember. most dropped right there, but I did have to fish a couple out of the water where they made a plunge before dying.
Hope this helps some?
PS:
DO NOT SHOOT THEM IN THE WATER!!
I'm sure you can figure why?