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For the following exercises, determine whether the statements are true or false .
If surface S is given by $\left\{\left(x,y,z\right):0\le x\le 1,0\le y\le 1,z=10\right\},$ then ${\iint}_{S}f\left(x,y,z\right)}dS={\displaystyle {\int}_{0}^{1}{\displaystyle {\int}_{0}^{1}f\left(x,y,10\right)}}dxdy.$
True
If surface S is given by $\left\{\left(x,y,z\right):0\le x\le 1,0\le y\le 1,z=x\right\},$ then ${\iint}_{S}f\left(x,y,z\right)}dS={\displaystyle {\int}_{0}^{1}{\displaystyle {\int}_{0}^{1}f\left(x,y,x\right)}}dxdy.$
Surface $\text{r}=\u27e8v\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u,v\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u,{v}^{2}\u27e9,\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}0\le u\le \pi ,0\le v\le 2,$ is the same as surface $\text{r}=\u27e8\sqrt{v}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}2u,\sqrt{v}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}2u,v\u27e9,$ for $0\le u\le \frac{\pi}{2},0\le v\le 4.$
True
Given the standard parameterization of a sphere, normal vectors ${\text{t}}_{u}^{}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{\text{t}}_{v}$ are outward normal vectors.
For the following exercises, find parametric descriptions for the following surfaces.
Plane $3x-2y+z=2$
$\text{r}\left(u,v\right)=\u27e8u,v,2-3u+2v\u27e9$ for $\text{\u2212}\infty \le u<\infty $ and $\text{\u2212}\infty \le v<\infty .$
Paraboloid $z={x}^{2}+{y}^{2},$ for $0\le z\le 9.$
Plane $2x-4y+3z=16$
$\text{r}(u,v)=\u27e8u,v,\frac{1}{3}\left(16-2u+4v\right)\u27e9$ for $\left|u\right|<\infty $ and $\left|v\right|<\infty .$
The frustum of cone ${z}^{2}={x}^{2}+{y}^{2},\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}2\le z\le 8$
The portion of cylinder ${x}^{2}+{y}^{2}=9$ in the first octant, for $0\le z\le 3$
$\text{r}(u,v)=\u27e83\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u,3\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u,v\u27e9$ for $0\le u\le \frac{\pi}{2},0\le v\le 3$
A cone with base radius r and height h , where r and h are positive constants
For the following exercises, use a computer algebra system to approximate the area of the following surfaces using a parametric description of the surface.
[T] Half cylinder $\left\{\left(r,\theta ,z\right):r=4,0\le \theta \le \pi ,0\le z\le 7\right\}$
$A=87.9646$
[T] Plane $z=10-x-y$ above square $\left|x\right|\le 2,\left|y\right|\le 2$
For the following exercises, let S be the hemisphere ${x}^{2}+{y}^{2}+{z}^{2}=4,$ with $z\ge 0,$ and evaluate each surface integral, in the counterclockwise direction.
${\iint}_{S}(x-2y)dS$
${\iint}_{S}\left({x}^{2}+{y}^{2}\right)zdS$
${\iint}_{S}\left({x}^{2}+{y}^{2}\right)zdS}=16\pi $
For the following exercises, evaluate $\int {\displaystyle {\int}_{S}\text{F}\xb7\text{N}ds}$ for vector field F , where N is an outward normal vector to surface S.
$\text{F}\left(x,y,z\right)=x\text{i}+2y\text{j}=3z\text{k},$ and S is that part of plane $15x-12y+3z=6$ that lies above unit square $0\le x\le 1,0\le y\le 1.$
$\text{F}(x,y)=x\text{i}+y\text{j},$ and S is hemisphere $z=\sqrt{1-{x}^{2}-{y}^{2}}.$
$\iint}_{S}\text{F}\xb7\text{N}dS=\frac{4\pi}{3$
$\text{F}(x,y,z)={x}^{2}\text{i}+{y}^{2}\text{j}+{z}^{2}\text{k},$ and S is the portion of plane $z=y+1$ that lies inside cylinder ${x}^{2}+{y}^{2}=1.$
For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places.
[T] S is surface $z=4-x-2y,\phantom{\rule{0.2em}{0ex}}\text{with}\phantom{\rule{0.2em}{0ex}}z\ge 0\text{,}\phantom{\rule{0.2em}{0ex}}x\ge 0\text{,}\phantom{\rule{0.2em}{0ex}}y\ge 0\text{;}\phantom{\rule{0.2em}{0ex}}\xi =x.$
$m\approx 13.0639$
[T] S is surface $z={x}^{2}+{y}^{2},\phantom{\rule{0.2em}{0ex}}\text{with}\phantom{\rule{0.2em}{0ex}}z\le 1\text{;}\phantom{\rule{0.2em}{0ex}}\xi =z.$
[T] S is surface ${x}^{2}+{y}^{2}+{x}^{2}=5,\phantom{\rule{0.2em}{0ex}}\text{with}\phantom{\rule{0.2em}{0ex}}z\ge 1\text{;}\phantom{\rule{0.2em}{0ex}}\xi ={\theta}^{2}.$
$m\approx 228.5313$
Evaluate ${\iint}_{S}\left({y}^{2}z\text{i}+{y}^{3}\text{j}+xz\text{k}\right)\xb7d\text{S}}\text{,$ where S is the surface of cube $\mathrm{-1}\le x\le 1,\mathrm{-1}\le y\le 1,\text{and}\phantom{\rule{0.2em}{0ex}}0\le z\le 2.$ in a counterclockwise direction.
Evaluate surface integral ${\iint}_{S}gdS},$ where $g(x,y,z)=xz+2{x}^{2}-3xy$ and S is the portion of plane $2x-3y+z=6$ that lies over unit square R : $0\le x\le 1\text{,}\phantom{\rule{0.2em}{0ex}}0\le y\le 1.$
${\iint}_{S}gdS}=3\sqrt{4$
Evaluate ${\iint}_{S}(x+y+z)d\text{S}}\text{,$ where S is the surface defined parametrically by $\text{R}(u,v)=(2u+v)\text{i}+(u-2v)\text{j}+(u+3v)\text{k}$ for $0\le u\le 1,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}0\le v\le 2.$
[T] Evaluate ${\iint}_{S}(x-{y}^{2}+z)d\text{S}}\text{,$ where S is the surface defined by $\text{R}(u,v)={u}^{2}\text{i}+v\text{j}+u\text{k}\text{,}\phantom{\rule{0.2em}{0ex}}0\le u\le 1\text{,}\phantom{\rule{0.2em}{0ex}}0\le v\le 1.$
${\iint}_{S}\left({x}^{2}+y-z\right)}d\text{S}\approx 0.9617$
[T] Evaluate where S is the surface defined by $\text{R}(u,v)=u\text{i}-{u}^{2}\text{j}+v\text{k}\text{,}\phantom{\rule{0.2em}{0ex}}0\le u\le 2\text{,}\phantom{\rule{0.2em}{0ex}}0\le v\le 1.$ for $0\le u\le 1\text{,}\phantom{\rule{0.2em}{0ex}}0\le v\le 2.$
Evaluate ${\iint}_{S}\left({x}^{2}+{y}^{2}\right)d\text{S}},$ where S is the surface bounded above hemisphere $z=\sqrt{1-{x}^{2}-{y}^{2}},$ and below by plane $z=0.$
${\iint}_{S}\left({x}^{2}+{y}^{2}\right)d\text{S}}=\frac{4\pi}{3$
Evaluate ${\iint}_{S}\left({x}^{2}+{y}^{2}+{z}^{2}\right)d\text{S}},$ where S is the portion of plane $z=x+1$ that lies inside cylinder ${x}^{2}+{y}^{2}=1.$
[T] Evaluate ${\iint}_{S}{x}^{2}zdS,$ where S is the portion of cone ${z}^{2}={x}^{2}+{y}^{2}$ that lies between planes $z=1$ and $z=4.$
${\iint}_{S}{x}^{2}zd\text{S}}=\frac{1023\sqrt{2\pi}}{5$
[T] Evaluate ${\iint}_{S}\left(xz\text{/}y\right)dS},$ where S is the portion of cylinder $x={y}^{2}$ that lies in the first octant between planes $z=0,z=5,y=1,$ and $y=4.$
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