### Surface Integrals

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1. Evaluate the surface integral $\underset{S}{\iint }{x}^{2}yz\phantom{\rule{0.2em}{0ex}}dS$ if $S$ is the part of the plane $z=1+2x+3y$ above the rectangle $\left[0,3\right]×\left[0,2\right]$.
$171\sqrt{14}$
2. Evaluate the surface integral $\underset{S}{\iint }yz\phantom{\rule{0.2em}{0ex}}dS$ if $S$ is the part of the plane $x+y+z=1$ that lies in the first octant.
$\frac{\sqrt{3}}{24}$
3. Evaluate the surface integral $\underset{S}{\iint }{x}^{2}{z}^{2}\phantom{\rule{0.2em}{0ex}}dS$ if $S$ is the part of the cone ${z}^{2}={x}^{2}+{y}^{2}$ that lies between the planes $z=1$ and $z=3$.
$\frac{364\sqrt{2}\pi }{3}$
4. Evaluate the surface integral $\underset{S}{\iint }y\phantom{\rule{0.2em}{0ex}}dS$ if $S$ is the part of the paraboloid $y={x}^{2}+{z}^{2}$ that lies inside the cylinder ${x}^{2}+{z}^{2}=4$.
$\frac{\pi \left(391\sqrt{17}+1\right)}{60}$
5. Evaluate the surface integral $\underset{S}{\iint }\left({x}^{2}z+{y}^{2}z\right)\phantom{\rule{0.2em}{0ex}}dS$ if $S$ is the part of the hemisphere ${x}^{2}+{y}^{2}+{z}^{2}=4$ above the $xy\text{-plane}$.
$16\pi$
6. Find the flux of $\mathbf{F}$ across $S$ if $\mathbf{F}\left(x,y,z\right)=⟨xy,yz,zx⟩$ and $S$ is the part of the paraboloid $z=4-{x}^{2}-{y}^{2}$ that lies above the square $0\le x\le 1$, $0\le y\le 1$, and has upward orientation.
$\frac{713}{180}$
7. Find the flux of $\mathbf{F}$ across $S$ if $\mathbf{F}\left(x,y,z\right)=⟨x,-z,y⟩$ and $S$ is the part of the sphere ${x}^{2}+{y}^{2}+{z}^{2}=4$ in the first octant, oriented towards the origin.
$-\frac{4}{3}\pi$
8. Find the flux of $\mathbf{F}$ across $S$ if $\mathbf{F}\left(x,y,z\right)=⟨0,y,-z⟩$ and $S$ is a positively oriented surface which consists of the paraboloid $y={x}^{2}+{z}^{2}$, $0\le y\le 1$, and the disk ${x}^{2}+{z}^{2}\le 1$, $y=1$.
$0$
9. Find the centroid of the hemisphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2}$, $z\ge 0$.
$\left(0,0,a/2\right)$
10. A fluid has density $870$ $\mathrm{kg}/{\mathrm{m}}^{3}$ and flows with velocity $\mathbf{v}=z\mathbf{i}+{y}^{2}\mathbf{j}+{x}^{2}\mathbf{k}$, where $x$, $y$, and $z$ are measured in meters and the components of $\mathbf{v}$ are measured in meters per second. Find the rate of flow outward through the cylinder ${x}^{2}+{y}^{2}=4$, $0\le z\le 1$.
$0$ $\frac{\mathrm{kg}}{\mathrm{s}}$