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Multiple Choice If a surface \(S\) has a unit normal vector \(\mathbf{n}\) at each point of the surface that can be chosen so \(\mathbf{n}\) varies continuously over the surface, and as \(\mathbf{n} \) moves around any closed curve on the surface, \(\mathbf{n}\) returns to its original direction, then the surface is said to be [(a) a Möbius strip, (b) integrable, (c) orientable].

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Multiple Choice  If \(\mathbf{F}=\mathbf{F}(x,y,z)\) is a vector field that represents the velocity of a fluid, and \(\rho =\rho (x,y,z)\) is the variable mass density of the fluid, then the surface integral \(\iint\limits_{\kern-3ptS}\rho \mathbf{F}\,{\cdot}\, \mathbf{n}\,dS\) is called the [(a) speed, (b) flux, (c) orientation, (d) acceleration] of \(\mathbf{F}\) across the surface \(S\).

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\(\iint\limits_{\kern-3ptS}(x^{2}+y^{2})\,dS;\quad S: z=f(x,y)=2x+3y+5, \ 0\leq x\leq 1, 0\leq y\leq 1\)

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\(\iint\limits_{\kern-3ptS}x^{2}y^{2}\,dS;\quad S: z=f(x,y)=3x+4y+8, 0\leq x\leq 1, 0\leq y\leq 1\)

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\(\iint\limits_{\kern-3ptS}y\,dS;\quad S: z=f(x,y)=x+5y^{2}, 0\leq x\leq2, 0\leq y\leq1\)

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\(\iint\limits_{\kern-3ptS}4x\,dS;\quad S: z=f(x,y)=2x^{2}+y, 0\leq x\leq1, 0\leq y\leq2\)

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\(\iint\limits_{\kern-3ptS}(x+y+z)\,dS\), where \(S\) is the portion of the plane \(z=x-3y\) above the region enclosed by \(y=0\), \(x=1\), and \(x=3y\) in the \(xy\)-plane

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\(\iint\limits_{\kern-3ptS}x\,dS\), where \(S\) is the portion of the plane \(x+y+2z=4\) above the region enclosed by \(x=1\), \(y=1\), \(x=0\), and \(y=0\)

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\(\iint\limits_{\kern-3ptS}y\,dS\), where \(S\) is the portion of the plane \(x+y+z=2\) inside the cylinder \(x^{2}+y^{2}=1\)

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\(\iint\limits_{\kern-3ptS}yz\,dS\), where \(S\) is the portion of the plane \(x+2y+3z=6\) in the first octant

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\(\iint\limits_{\kern-3ptS}x^{2}z\,dS\), where \(S\) is the surface \(x^{2}+y^{2}=1\), \(0\leq z\leq 1\)

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\(\iint\limits_{\kern-3ptS}(x+y)z\,dS,\) where \(S\) is the surface \(x^{2} + y^{2} = 9, 1\leq z \leq 3\)

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\(z=f(x,y)=\sqrt{16-x^{2}-y^{2}},\quad 0\leq x^{2}+y^{2}\leq 16\)

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\(z=f(x,y)=\sqrt{1-x^{2}-y^{2}},\quad 0\leq x^{2}+y^{2}\leq 1\)

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\(z=f(x,y)=\sqrt{36-9x^{2}-4y^{2}},\quad 0\leq 9x^{2}+4y^{2}\leq 36 \)

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\(z=f(x,y)=\sqrt{4-x^{2}-4y^{2}},\quad 0\leq x^{2}+4y^{2}\leq 4 \)

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\(S: \mathbf{r}(u,v)=(u+v)\mathbf{i}+u^{2}\mathbf{j}+v^{2}\mathbf{k}\)

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\(S: \mathbf{r}(u,v)=uv\mathbf{i}+v\mathbf{j}+e^{u}\mathbf{k}\)

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\(\mathbf{F}=x\mathbf{i}\)

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\(\mathbf{F}=y\mathbf{i}\)

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\(\mathbf{F}=z\mathbf{i}\)

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\(\mathbf{F}=x\mathbf{i}+y\mathbf{j}\)

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\(\mathbf{F}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\)

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\(\mathbf{F}=z^{2}\mathbf{i}\)

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\(\mathbf{F}=x^{2}\mathbf{i}+y^{2}\mathbf{j}+z^{2}\mathbf{k}\)

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\(\mathbf{F}=x^{2}\mathbf{i}+y^{2}\mathbf{j}\)

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\(\mathbf{F}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\), and \(S\) is the surface \(x^{2}+y^{2}+z^{2}=1\), \(z\geq 0\)

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\(\mathbf{F}=-y\mathbf{i}+x\mathbf{j}+z\mathbf{k}\), and \(S\) is the surface \(x^{2}+y^{2}+z^{2}=1\), \(z\geq 0\)

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\(\mathbf{F}=(x+y)\mathbf{i}+(2x-z)\mathbf{j}+y\mathbf{k}\), and \(S\) is the tetrahedron formed by the coordinate planes and the plane \( z+2x+2y=8\)

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\(\mathbf{F}=2x\mathbf{i}-x^{2}\mathbf{j}+(z-2x+2y)\mathbf{k}\) , and \(S\) is the tetrahedron formed by the coordinate planes and the plane \(2x+2y+z=6\)

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The surface \(S\) is parametrized by \(\mathbf{r}(u,v)=uv\,\mathbf{i}+u^{2}\mathbf{j}+v\,\mathbf{k}\), \(1\leq u\leq 2\), \(0\leq v\leq 3 \) and the vector field is \(\mathbf{F}=\mathbf{F}(x,y,z)=-z\mathbf{j}+\mathbf{k}\).

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The surface \(S\) is parametrized by \(\mathbf{r}(u,v)=ve^{u} \mathbf{i}+v^{3}\mathbf{j}+u\mathbf{k}\), \(0\leq u\leq 1\), \(-1\leq v\leq 1\) and the vector field is \(\mathbf{F}=\mathbf{F}(x,y,z)=xy\mathbf{i}+2z\mathbf{ j}\).

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Mass of a Lamina Find the mass of a lamina in the shape of a cone \(z=\sqrt{x^{2}+y^{2}}\), \(2\leq z\leq 4\), if the mass density of the lamina is \(\rho =\rho (x,y,z)=8-z\).

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Mass of a Lamina Find the mass of a lamina in the shape of a cone \(z=\sqrt{x^{2}+y^{2}}\), \(2\leq z\leq 4\), if the mass density of the lamina is \(\rho =\rho (x,y,z)=\sqrt{x^{2}+y^{2}}\).

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Center of Mass of a Lamina Find the center of mass of a lamina in the shape of the part of the plane \(2x+3y+z=4\) that lies inside \(x^{2}+z^{2}=9\), if the mass density of the lamina is \(\rho =\rho (x,y,z)= \sqrt{x^{2}+z^{2}}\).

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Center of Mass of a Lamina Find the center of mass of a lamina in the shape of the unit sphere \(x^{2}+y^{2}+z^{2}=1\), if the mass density of the lamina is \(\rho =\rho (x,y,z)=z+1\).

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\(\iint\limits_{\kern-3ptS}x^{2}z\,dS;\quad S\): the portion of the cylinder \(x^{2}+y^{2}=9\) between \(z=0\) and \(z=5\)

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\(\iint\limits_{\kern-3ptS}(x+2y)\,dS ;\quad S: \mathbf{r}(u,v)=u\cos v\,\mathbf{i}+(u^{2}+1)\mathbf{j}+u\cos v\,\mathbf{k}\), \(0\leq u\leq 1\) and \(0\leq v\leq \pi /2\)

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Flux Across a Surface Find the mass of a fluid with constant mass density flowing across the paraboloid \(z=x^{2}+y^{2}\), \(z\leq 5 \), in a unit of time in the direction of the outer unit normal, if the velocity of the fluid at any point on the paraboloid is \(\mathbf{F}=\mathbf{F }(x,y,z)=-x\mathbf{i}-y\mathbf{j}-z\mathbf{k}\).

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Flux Across a Surface  Find the mass of a fluid with constant mass density flowing across the paraboloid \(z=9-x^{2}-y^{2}\), \(z\geq 0\), in a unit of time in the direction of the outer unit normal, if the velocity of the fluid at any point on the paraboloid is \(\mathbf{F}=\mathbf{F} (x,y,z)=x\mathbf{i}+y\mathbf{j}+3\mathbf{k}\).

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If \(\mathbf{n}\) is the upward-pointing unit normal to a surface \( z=f(x,y)\) and \(\mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}\), show that \[ \iint\limits_{\kern-3ptS}\mathbf{F}\,{\cdot}\, \mathbf{n}\,dS=\iint\limits_{\kern-3ptD}\left( -P \dfrac{\partial f}{\partial x}-Q\dfrac{\partial f}{\partial y}+R\right) dx\,dy \]

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Electric Flux Find the electric flux across the part of the cylinder \(x^{2}+y^{2}=9,\) \(0\leq z\leq 2,\) in the direction of the outer unit normal vectors when the electric field is \(\mathbf{E}(x,y,z)=x\mathbf{i} +2y\mathbf{j}+3z\mathbf{k}\).

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Helicoid Find \(\iint\limits_{\kern-3ptS}(x^2+z^2)dS\), where \(S\) is the helicoid \(\mathbf{r}(u,v)=u\cos(2v){\bf i} + u\sin (2v){\bf j} + v{\bf k},0\leq u\leq 3, 0\leq v\leq 2\pi\).

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Center of Mass Find the center of mass of the upper half of the sphere \(x^{2}+y^{2}+z^{2}=a^{2}\) covered by a thin material with mass density at each point proportional to the distance from the \(xy\)-plane.

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Find \(\iint\limits_{\kern-3ptS}\mathbf{F}\,{\cdot}\, \mathbf{n}\,dS\), where \(S\) is a level surface of a function \(w= f(x,y,z)\) with outer unit normal vector \(\mathbf{n},\) \(\ \mathbf{F}={\nabla} f\), and \(f\) satisfies the equation \(\left( \dfrac{\partial f}{\partial x}\right) ^{2}+\left( \dfrac{\partial f}{ \partial y}\right) ^{2}+\left( \dfrac{\partial f}{\partial z}\right) ^{2}=1\). ( Hint: The answer should depend on \(S\).)