Solve for y:

We need to keep the "+" solution else y isn't real. So:

Thus

The area you are after will be between the "+" and "-" curves, and you know by symmetry they cross on the x-axis so you should be able to find these fairly easily. The integral you will be doing will be of the form:

which isn't going to be at all pretty. You'll likely have to do a numerical estimate. (Unless Krizalid or someone else pulls a trick out of their hat.)

-Dan