$\displaystyle \int\limits_C \vec{F} . dr $

C: $\displaystyle x^2 + y^2 + z^2 = a^2$ intersection $\displaystyle y + z = a$

$\displaystyle \vec{F} = (3y +z, x + 4y, 2x + y)$

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$\displaystyle \nabla_X\vec {F} = (1, -1, -2)$

$\displaystyle \vec{r(u,v)} = (u, v, a-v)$

$\displaystyle \vec{n} = (0, 1, 1)$

I made the intersection between the sphere and the plane and got:

$\displaystyle \frac{x^2}{a^2} + \frac{(y-a)^2}{\frac{a^2}{2}} = 1 $

I know the area of this ellipse, but what is the area of this ellipse projected on the xy plane?

If I knew this could substitute an got the value:

$\displaystyle \iint\limits_D -3 dudv = -3A(D)$