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**angel.white** Evaluate the surface integral $\displaystyle \int \int_S \vec F \cdot d \vec S$ for the given vector field $\displaystyle \vec F$ and the oriented surface S. In other words, find the flux of $\displaystyle \vec F$ across S. For closed surfaces, use the positive (outward orientation).

$\displaystyle \vec F(x,y,z) = y \vec j - z \vec k$,

S consists of the paraboloid $\displaystyle y=x^2+z^2$, $\displaystyle 0 \leq y \leq 1$

and the disk $\displaystyle x^2 + z^2 \leq 1$, y=1

I've thought about this for a bit, and I think it's a sideways half-circle with the base at y=1 and the apex at the origin. The correct answer is zero, I tried splitting it into two separate equations and adding their answers, which came out very nicely, but was not correct. I'm not sure how to set it up properly.