면적분과 플럭스

For the following exercises, determine whether the statements are true or false.

If surface S is given by $\{(x,y,z):0\le x\le 1,0\le y\le 1,z=10\},$ then ${\iint }_{S}f(x,y,z)dS={\int }_{0}^{1}{\int }_{0}^{1}f(x,y,10)dxdy.$

For the following exercises, determine whether the statements are true or false.

If surface S is given by $\{(x,y,z):0\le x\le 1,0\le y\le 1,z=x\},$ then ${\iint }_{S}f(x,y,z)dS={\int }_{0}^{1}{\int }_{0}^{1}f(x,y,x)dxdy.$

For the following exercises, determine whether the statements are true or false.

Surface $\text{r}=\langle v\,\cos \,u,v\,\sin \,u,{v}^{2}\rangle ,\,\text{for}\,0\le u\le \pi ,0\le v\le 2,$ is the same as surface $\text{r}=\langle \sqrt{v}\,\cos \,2u,\sqrt{v}\,\sin \,2u,v\rangle ,$ for $0\le u\le \frac{\pi }{2},0\le v\le 4.$

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

The portion of cylinder ${x}^{2}+{y}^{2}=9$ in the first octant, for $0\le z\le 3$

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 $\{(r,\theta ,z):r=4,0\le \theta \le \pi ,0\le z\le 7\}$

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] 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.

${\iint }_{S}zdS$

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 $z=\sqrt{1-{x}^{2}-{y}^{2}}.$

$\text{F}(x,y,z)={x}^{2}\text{i}+{y}^{2}\text{j}+{z}^{2}\text{k},$ 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 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 $z={x}^{2}+{y}^{2},\,\text{with}\,z\le 1;\,\rho =z.$