면적분과 플럭스

Find the mass of a lamina of density $\rho (x,y,z)=z$ in the shape of hemisphere $z={({a}^{2}-{x}^{2}-{y}^{2})}^{1/2}.$

Compute ${\iint }_{S}\text{F}\cdot \text{N}dS,$ where $\text{F}(x,y,z)=xy\text{i}+z\text{j}+(x+y)\text{k}$ and N is an upward pointing normal vector $S$, where S is the triangular region of the plane $x+y+z=1$ in the first octant.

Compute ${\iint }_{S}\text{F}\cdot \text{N}dS,$ where $\text{F}(x,y,z)=2yz\text{i}+({\tan }^{-1}(xz))\text{j}+{e}^{xy}\text{k}$ and N is an outward normal vector of S, where S is the surface of sphere ${x}^{2}+{y}^{2}+{z}^{2}=1.$

Compute ${\iint }_{S}\text{F}\cdot \text{N}dS,$ where $\text{F}(x,y,z)=xyz\text{i}+xyz\text{j}+xyz\text{k}$ and N is an outward normal vector S, where S is the surface of the five faces of the unit cube $0\le x\le 1,\,0\le y\le 1,\,0\le z\le 1$ missing $z=0.$

For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the yz-plane.

${\iint }_{S}x{y}^{2}{z}^{3}dS;$ S is the first-octant portion of plane $2x+3y+4z=12.$

For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz-plane

${\iint }_{S}x{y}^{2}{z}^{3}dS;$ S is the first-octant portion of plane $2x+3y+4z=12.$

Evaluate surface integral ${\iint }_{S}yzdS,$ where S is the first-octant part of plane $x+y+z=\lambda ,$ where $\lambda$ is a positive constant.

For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz-plane

Evaluate surface integral ${\iint }_{S}({x}^{2}z+{y}^{2}z)dS,$ where S is hemisphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2},z\ge 0.$

For the following exercises, express the surface integral as an iterated double integral by using a projection on S on the xz-plane

Evaluate surface integral ${\iint }_{S}zdS,$ where S is surface $z=\sqrt{{x}^{2}+{y}^{2}},0\le z\le 2.$

Evaluate surface integral ${\iint }_{S}{x}^{2}yzdS,$ where S is the part of plane $z=1+2x+3y$ that lies above rectangle $0\le x\le 3\,\text{and}\,0\le y\le 2.$

Evaluate surface integral ${\iint }_{S}yzdS,$ where S is plane $x+y+z=1$ that lies in the first octant.

Evaluate surface integral ${\iint }_{S}yzdS,$ where S is the part of plane $z=y+3$ that lies inside cylinder ${x}^{2}+{y}^{2}=1.$

For the following exercises, use geometric reasoning to evaluate the given surface integrals.

${\iint }_{S}\sqrt{{x}^{2}+{y}^{2}+{z}^{2}}dS,$ where S is surface ${x}^{2}+{y}^{2}+{z}^{2}=4,z\ge 0$

${\iint }_{S}(x\text{i}+y\text{j})\cdot dS,$ where S is surface ${x}^{2}+{y}^{2}=4,1\le z\le 3,$ oriented with unit normal vectors pointing outward

${\iint }_{S}(z\text{k})\cdot dS,$ where S is disc ${x}^{2}+{y}^{2}\le 9$ on plane $z=4,$ oriented with unit normal vectors pointing upward

In Exercises 5–6, a surface $\mathcal{S}$ that represents a thin sheet of material with density $\delta$ is given. Find the mass of each thin sheet.

$\mathcal{S}$ is the plane $f(x,y)=x+y$ on $-2\leq x\leq 2$, $-3\leq y\leq 3$, with $\delta(x,y,z) = z$.

In Exercises 5–6, a surface $\mathcal{S}$ that represents a thin sheet of material with density $\delta$ is given. Find the mass of each thin sheet.

$\mathcal{S}$ is the unit sphere, with $\delta(x,y,z) = x+y+z+10$.

In Exercises 7–14, a surface $\mathcal{S}$ and a vector field $\vec F$ are given. Compute the flux of $\vec F$ across $\mathcal{S}$. (If $\mathcal{S}$ is not a closed surface, choose $\vec n$ so that it has a positive $z$-component, unless otherwise indicated.)

$\mathcal{S}$ is the plane $f(x,y) = 3x+y$ on $0\leq x\leq 1$, $1\leq y\leq 4$; $\vec F = \langle x^2,-z,2y\rangle$.

In Exercises 7–14, a surface $\mathcal{S}$ and a vector field $\vec F$ are given. Compute the flux of $\vec F$ across $\mathcal{S}$. (If $\mathcal{S}$ is not a closed surface, choose $\vec n$ so that it has a positive $z$-component, unless otherwise indicated.)

$\mathcal{S}$ is the plane $f(x,y) = 8-x-y$ over the triangle with vertices at $(0,0)$, $(1,0)$ and $(1,5)$; $\vec F = \langle 3,1,2\rangle$.

In Exercises 7–14, a surface $\mathcal{S}$ and a vector field $\vec F$ are given. Compute the flux of $\vec F$ across $\mathcal{S}$. (If $\mathcal{S}$ is not a closed surface, choose $\vec n$ so that it has a positive $z$-component, unless otherwise indicated.)

$\mathcal{S}$ is the paraboloid $f(x,y) = x^2+y^2$ over the unit disk; $\vec F = \langle 1,0,0\rangle$.