Or, if line integral aren't your thing, you can do it with a basic cartesian equation of the ellipse.

Center the ellipse on the origin and write it as an equation, X^2/a^2 + Y^2/b^2 = 1. (where a is the X-axis radius and b is the y-axis radius) Solve for Y = f(x) and you have the equation for one half of the ellipse. Integrate the equation f(x) over the length in the x-axis and you have the area under the top half of the ellipse. Now double it, and there's your answer.

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