면적분과 플럭스

Given the standard parameterization of a sphere, normal vectors ${\text{t}}_{u}^{}\,\times \,{\text{t}}_{v}$ are outward normal vectors.

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

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

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

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

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

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

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

The frustum of cone ${z}^{2}={x}^{2}+{y}^{2},\,\text{for}\,2\le z\le 8$

A cone with base radius r and height h, where r and h are positive constants

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.

${\iint }_{S}zdS$

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.

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

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.

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

$\text{F}(x,y,z)=x\text{i}+2y\text{j}-3z\text{k},$ and S is that part of plane $15x-12y+3z=6$ that lies above unit square $0\le x\le 1,0\le y\le 1.$

For the following exercises, evaluate $\int {\int }_{S}\text{F}\cdot \text{N}dS$ for vector field F, where N is an upward pointing normal vector to surface S.

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

For the following exercises, approximate the mass of the lamina that has the shape of given surface S. Round to four decimal places.

[T] S is surface $z=4-x-2y,\,\text{with}\,z\ge 0,\,x\ge 0,\,y\ge 0;\,\rho =x.$

For the following exercises, approximate the mass of the lamina that has the shape of given surface S. Round to four decimal places.

[T] S is surface $z={x}^{2}+{y}^{2},\,\text{with}\,z\le 1;\,\rho =z.$

For the following exercises, approximate the mass of the lamina that has the shape of given surface S. Round to four decimal places.

[T] S is surface ${x}^{2}+{y}^{2}+{z}^{2}=5,\,\text{with}\,z\ge 1;\,\rho ={\theta }^{2}.$

Evaluate ${\iint }_{S}({y}^{2}z\text{i}+{y}^{3}\text{j}+xz\text{k})\cdot dS,$ where S is the surface of cube $-1\le x\le 1,-1\le y\le 1,\text{and}\,0\le z\le 2.$ Assume an outward pointing normal.

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: $0\le x\le 1,\,0\le y\le 1.$

For the following exercises, approximate the mass of the lamina that has the shape of given surface S. Round to four decimal places.

[T] Evaluate $\int {\int }_{S}({x}^{2}+{y}^{2}-z)dS$ where $S$ is the surface defined by $\text{r}(u,v)=u\text{i}-{u}^{2}\text{j}+v\text{k},\,0\le u\le 2,\,0\le v\le 1.$

Evaluate ${\iint }_{S}({x}^{2}+{y}^{2})dS,$ where S is the surface of hemisphere $z=\sqrt{1-{x}^{2}-{y}^{2}},$ and above the plane $z=0.$

Evaluate ${\iint }_{S}({x}^{2}+{y}^{2}+{z}^{2})dS,$ where S is the portion of plane $z=x+1$ that lies inside cylinder ${x}^{2}+{y}^{2}=1.$

Evaluate ${\iint }_{S}xyzdS$ if S is the part of plane $z=x+y$ that lies over the triangular region in the xy-plane with vertices (0, 0, 0), (1, 0, 0), and (0, 2, 0).