면적분과 플럭스

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

$\text{F}(x,y,z)={x}^{2}\text{i}+{y}^{2}\text{j}+{z}^{2}\text{k},$ and S is the portion of plane $z=y+1$ that lies inside cylinder ${x}^{2}+{y}^{2}=1.$

Evaluate ${\iint }_{S}({x}^{2}+y-z)dS,$ 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 $0\le u\le 1,\,\text{and}\,0\le v\le 2.$

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

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

[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 $z=4.$

[T] Evaluate ${\iint }_{S}(xz/y)dS,$ 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 $y=4.$

[T] Evaluate ${\iint }_{S}(z+y)dS,$ where S is the part of the graph of $z=\sqrt{1-{x}^{2}}$ in the first octant between the xz-plane and plane $y=3.$

Compute ${\iint }_{S}\text{F}\cdot \text{N}dS,$ where $\text{F}(x,y,z)=x\text{i}-5y\text{j}+4z\text{k}$ and N is an outward normal vector of S, where S is the union of two squares ${S}_{1}:x=0,\,0\le y\le 1,\,0\le z\le 1$ and ${S}_{2}:z=1,\,0\le x\le 1,\,0\le y\le 1.$

${\iint }_{S}({x}^{2}-2y+z)dS;$ S is the portion of the graph of $4x+y=8$ bounded by the coordinate planes and plane $z=6.$

${\iint }_{S}({x}^{2}-2y+z)dS;$ S is the portion of the graph of $4x+y=8$ bounded by the coordinate planes and plane $z=6.$

A lamina has the shape of a portion of sphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2}$ that lies within cone $z=\sqrt{{x}^{2}+{y}^{2}}.$ Let S be the spherical shell centered at the origin with radius a, and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Determine the mass of the lamina if $\rho (x,y,z)={x}^{2}{y}^{2}z.$

A lamina has the shape of a portion of sphere ${x}^{2}+{y}^{2}+{z}^{2}={a}^{2}$ that lies within cone $z=\cot {\varphi }_{0}\sqrt{{x}^{2}+{y}^{2}}$ Let S be the spherical shell centered at the origin with radius a, and let C be the right circular cone with a vertex at the origin and an axis of symmetry that coincides with the z-axis. Suppose the angle between the sides of the cone and the z-axis is ${\phi }_{0},\,\text{with}\,0\le {\phi }_{0}<\frac{\pi }{2}.$ Determine the mass of that portion of the shape enclosed in the intersection of S and C. Assume $\rho (x,y,z)={x}^{2}{y}^{2}z.$

For the following exercises, consider the radial fields $\text{F}=\frac{\langle x,y,z\rangle }{{({x}^{2}+{y}^{2}+{z}^{2})}^{\frac{p}{2}}}=\frac{\text{r}}{{||\text{r}||}^{p}},$ where p is a real number. Let S consist of spheres A and B centered at the origin with radii $0<a<b.$ The total outward flux across S consists of the outward flux across the outer sphere B less the flux into S across inner sphere A.

Find the total flux across S with $p=0.$

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 closed surface composed of $\mathcal{S}_1$, whose boundary is the ellipse in the $x$-$y$ plane described by $\frac{x^2}{25}+\frac{y^2}9 = 1$ and $\mathcal{S}_2$, part of the elliptical paraboloid $f(x,y) = 1-\frac{x^2}{25}-\frac{y^2}9$ (see graph); $\vec F = \langle 5,2,3\rangle$.

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 closed surface composed of $\mathcal{S}_1$, part of the unit sphere and $\mathcal{S}_2$, part of the plane $z=1/2$ (see graph); $\vec F = \langle x,-y,z\rangle$.