I don't think battery calibration is going to be an issue if it only lasts 10 minutes, although it can't hurt.

If that battery update thing hasn't been done on your machine, that is probably the first thing they'll have you do. But you never know. They may decide such a severe issue is not any of those typical problems and just give you a new one. There's also a recall out there that may or may not affect your battery serial number.

If the roughly a year is a little under, you should still be covered under warranty. Battery is a wear part, but if that's all the time you get from it, it should be covered. I'm about 9 months in on mine and was able to get a new battery recently even though mine ran for much longer than yours (but much shorter than it used to in the beginning).