Just note that the area of the slices will change:
Take note of this: To evaluate the integral...make a trig substutition:
Can you take it from here?
EDIT : Woops...I just noticed that the equation changed slightly...
the distance from one side of the ellipse to the other should now be (Verify)
Thus, the area of each slice would be and the volume would be
You still would need to apply a trig substitution: