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Surface integral over 2D slice of ellipsoid

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I am trying to take the surface integral on a 2D ellipsoidal slice. These 3D ellipsoids are contained within a cylinder. At first I tried to do a parameterized surface, however when I enter the parametric equations for the ellipse, it returns a shaded rectangular area. Then I tried to create a cut plane and use a boolean expression in the integral, but that didn't work either. Suggestions welcomed, thank you.


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