Does this sound like my laptop's battery is the problem?

I have a Vostro 3500 Windows 7 laptop from Dell with an intel core i5. My laptop has ran out of batteries faster and faster (staying charged for shorter amounts of time). If I leave it plugged in for too long it shuts down suddenly and unexpectedly, and if I run too many programs at once it also shuts down unexpectedly. I'm wondering if the problem is maybe that my laptop's battery is dying. Does that sound like that's the problem? Or could it be something else?

Thanks =)