# Calculate &#x222B;<!-- ∫ --> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="norm

Calculate
${\int }_{\mathrm{\partial }F}2xy\phantom{\rule{thinmathspace}{0ex}}dx+{x}^{2}\phantom{\rule{thinmathspace}{0ex}}dy+\left(1+x-z\right)\phantom{\rule{thinmathspace}{0ex}}dz$
for the intersection of $z={x}^{2}+{y}^{2}$ and $2x+2y+z=7$. Go clockwise with respect to the origin.
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Rebekah Zimmerman
You should consider the surface S given by the intersection of the cylinder $\left(x+1{\right)}^{2}+\left(y+1{\right)}^{2}\le 7+2=9$ with the plane $2x+2y+z=7$. Then $\mathbf{n}=-\left(2,2,1\right)/3$ (the ellipse $\mathrm{\partial }S$ is clockwise oriented) and by Stokes' theorem: