6.6 Surface integrals  (Page 14/27)

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Key equations

• Scalar surface integral
$\int {\int }_{S}f\left(x,y,z\right)dS=\int {\int }_{D}f\left(\text{r}\left(u,v\right)\right)||{\text{t}}_{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{\text{t}}_{v}||dA$
• Flux integral
${\iint }_{S}\text{F}·\text{N}dS={\iint }_{S}\text{F}·d\text{S}={\iint }_{D}\text{F}\left(\text{r}\left(u,v\right)\right)·\left({\text{t}}_{u}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{\text{t}}_{v}\right)dA$

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 ${\iint }_{S}f\left(x,y,z\right)dS={\int }_{0}^{1}{\int }_{0}^{1}f\left(x,y,10\right)dxdy.$

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 ${\iint }_{S}f\left(x,y,z\right)dS={\int }_{0}^{1}{\int }_{0}^{1}f\left(x,y,x\right)dxdy.$

Surface $\text{r}=⟨v\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u,v\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u,{v}^{2}⟩,\phantom{\rule{0.2em}{0ex}}\text{for}\phantom{\rule{0.2em}{0ex}}0\le u\le \pi ,0\le v\le 2,$ is the same as surface $\text{r}=⟨\sqrt{v}\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}2u,\sqrt{v}\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}2u,v⟩,$ for $0\le u\le \frac{\pi }{2},0\le v\le 4.$

True

Given the standard parameterization of a sphere, normal vectors ${\text{t}}_{u}^{}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{\text{t}}_{v}$ are outward normal vectors.

For the following exercises, find parametric descriptions for the following surfaces.

Plane $3x-2y+z=2$

$\text{r}\left(u,v\right)=⟨u,v,2-3u+2v⟩$ for $\text{−}\infty \le u<\infty$ and $\text{−}\infty \le v<\infty .$

Paraboloid $z={x}^{2}+{y}^{2},$ for $0\le z\le 9.$

Plane $2x-4y+3z=16$

$\text{r}\left(u,v\right)=⟨u,v,\frac{1}{3}\left(16-2u+4v\right)⟩$ for $|u|<\infty$ and $|v|<\infty .$

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 $0\le z\le 3$

$\text{r}\left(u,v\right)=⟨3\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}u,3\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}u,v⟩$ 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 $|x|\le 2,|y|\le 2$

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.

${\iint }_{S}zdS$

${\iint }_{S}zdS=8\pi$

${\iint }_{S}\left(x-2y\right)dS$

${\iint }_{S}\left({x}^{2}+{y}^{2}\right)zdS$

${\iint }_{S}\left({x}^{2}+{y}^{2}\right)zdS=16\pi$

For the following exercises, evaluate $\int {\int }_{S}\text{F}·\text{N}ds$ 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 $0\le x\le 1,0\le y\le 1.$

$\text{F}\left(x,y\right)=x\text{i}+y\text{j},$ and S is hemisphere $z=\sqrt{1-{x}^{2}-{y}^{2}}.$

${\iint }_{S}\text{F}·\text{N}dS=\frac{4\pi }{3}$

$\text{F}\left(x,y,z\right)={x}^{2}\text{i}+{y}^{2}\text{j}+{z}^{2}\text{k},$ 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 ${\iint }_{S}\left({y}^{2}z\text{i}+{y}^{3}\text{j}+xz\text{k}\right)·d\text{S}\text{,}$ where S is the surface of cube $-1\le x\le 1,-1\le y\le 1,\text{and}\phantom{\rule{0.2em}{0ex}}0\le z\le 2.$ in a counterclockwise direction.

Evaluate surface integral ${\iint }_{S}gdS,$ where $g\left(x,y,z\right)=xz+2{x}^{2}-3xy$ 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 ${\iint }_{S}\left(x+y+z\right)d\text{S}\text{,}$ where S is the surface defined parametrically by $\text{R}\left(u,v\right)=\left(2u+v\right)\text{i}+\left(u-2v\right)\text{j}+\left(u+3v\right)\text{k}$ 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 ${\iint }_{S}\left(x-{y}^{2}+z\right)d\text{S}\text{,}$ where S is the surface defined by $\text{R}\left(u,v\right)={u}^{2}\text{i}+v\text{j}+u\text{k}\text{,}\phantom{\rule{0.2em}{0ex}}0\le u\le 1\text{,}\phantom{\rule{0.2em}{0ex}}0\le v\le 1.$

${\iint }_{S}\left({x}^{2}+y-z\right)d\text{S}\approx 0.9617$

[T] Evaluate where S is the surface defined by $\text{R}\left(u,v\right)=u\text{i}-{u}^{2}\text{j}+v\text{k}\text{,}\phantom{\rule{0.2em}{0ex}}0\le u\le 2\text{,}\phantom{\rule{0.2em}{0ex}}0\le v\le 1.$ 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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