The ellipse is centered at the origin, stretched in the y-direction, and intersects with the vertical line at:
Solving for x from the ellipse's upper half, in terms of y
The equation above represents the curve to the right. The curve to the left is simply:
Setup the integral now as
INTEGRAL (X_right - X_left) dy where y ranges from 0 to
After you find the integral remember to multiply it by two to account for the other half of the area below the horizontal axis.
I hope this helps.