if this is happening while you are plugged into AC, then I'd say it's unlikely to be the battery alone.
about the battery itself - select About This Mac in the Apple menu, then click on More Info, this will run System Profiler. On the left side, under the Hardware section, select "Power" and you can check info about your battery's condition on the right side of the window. for example see what it says beside Condition and # of cycles. more than about 300 cycles is considered the normal lifespan of the battery, according to Apple. a dying battery may report its condition as "Check battery". with my MBP, over 3 yrs I've had two batteries die prematurely (<300 cycles) - in one case, when running on battery power, the machine would suddenly shut down without warning at 40% remaining charge. AppleCare replaced both my batteries.
but I've not had a shutdown while plugged in like you report. I'd guess you may have some other issue going on, even if your battery is dead/dying. perhaps a flakey AC adapter?? do you have a friend with a MBP AC adapter that you can borrow to try?