Oh yes, the core is very important.
I talk about it here all the time, going on and on about the importance of Swiss balls and balance boards in core training.
Why, the core is absolutely essential to strength, health, and long life.
My advice is to get your one-legged Swiss ball squat up to 95 lbs. as quickly as possible so that your fucking core will be stable.
It's by far the best core exercise you can fucking do.