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I am trying to evaluate by a method "most suitable": $$\int_{C} F(r)\cdot dr $$

where $$F=[x^3, e^{2y}, e^{-yz}], C: x^2+9y^2 = 9, z=x^2$$

In the xy-plane this looks like an ellipse (I think), and it's a parabola in the xz-plane.

I am trying to parametrize this, so I was thinking of using polar coordinates, but I am getting thrown off by the $z=x^2$.

I am having a tough time thinking this one through. Maybe I need to use Stokes'?

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    The curve isn't an elliptic paraboloid at all (this is a surface). It is the intersection of an elliptic cylindre with a parabolic cylindre. – 2017-02-11

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You can try with

$$x=3\cos t, y=\sin t, z=9\cos^2t.$$