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**TheEmptySet** We can paramterize the surface by

$\displaystyle \vec r(u,v)=< 3\sin(v)\cos(u) ,3\sin(v)\sin(u),3\cos(v)>$

$\displaystyle \vec r_u(u,v)=<-3\sin(v)\sin(u),3\sin(v)\cos(u),0>$

$\displaystyle \vec r_v(u,v)=<3\cos(v)\cos(u),3\cos(v)\sin(u),-3\sin(v)>$

Note that u ranges from 0 to $\displaystyle 2\pi$

and v $\displaystyle 2=3\cos(v_0) \iff v_0 = \cos^{-1}\left( \frac{2}{3}\right)$

$\displaystyle 1=3\cos(v_1) \iff v_1 = \cos^{-1}\left( \frac{1}{3}\right)$

Now

$\displaystyle \iint_S \vec{F} \cdot d\vec{S}=\int_{0}^{2\pi} \int_{\cos^{-1}(1/3)}^{\cos^{-1}(2/3)}\vec{F}\cdot (r_u \times r_v)dudv$

This should get you started.