벡터 미적분 전체 문제

[T] Evaluate line integral ${\int }_{\gamma }^{}({y}^{2}-xy)ds,$ where $\gamma$ is curve $y=\ln \,x$ from (1, 0) toward $(e,\,1).$

[T] Evaluate line integral ${\int }_{\gamma }^{}x{y}^{4}ds,$ where $\gamma$ is the right half of circle ${x}^{2}+{y}^{2}=16.$

For the following exercises, use a CAS to evaluate the given line integrals.

[T] Evaluate ${\int }_{C}^{}\text{F}\cdot d\text{r},$ where $\text{F}(x,y,z)={x}^{2}y\text{i}+(x-z)\text{j}+xyz\text{k}$ and

C: $\text{r}(t)=t\text{i}+{t}^{2}\text{j}+2\text{k},\,0\le t\le 1.$

Evaluate ${\int }_{C}^{}\text{F}\cdot d\text{r},$ where $\text{F}(x,y)=2x\,\sin (y)\text{i}+({x}^{2}\cos (y)-3{y}^{2})\text{j}$ and

C is any path from $(-1,0)$ to (5, 1).

Find the line integral of $\text{F}(x,y,z)=12{x}^{2}\text{i}-5xy\text{j}+xz\text{k}$ over path C defined by $y={x}^{2},$ $z={x}^{3}$ from point (0, 0, 0) to point (2, 4, 8).

Find the line integral of ${\int }_{C}^{}(1+{x}^{2}y)ds,$ where C is ellipse $\text{r}(t)=2\,\cos \,t\text{i}+3\,\sin \,t\text{j}$ from $0\le t\le \pi .$

Compute the flux of $\text{F}={x}^{2}\text{i}+y\text{j}$ across a line segment from (0, 0) to (1, 2).

Let $\text{F}=5\text{i}$ and let C be curve $y=0,0\le x\le 4.$ Find the flux across C.

Let $\text{F}=5\text{j}$ and let C be curve $y=0,0\le x\le 4.$ Find the flux across C.

For the following exercises, find the flux.

Let $\text{F}=-y\text{i}+x\text{j}$ and let C: $\text{r}(t)=\cos \,t\text{i}+\sin \,t\text{j}$ $(0\le t\le 2\pi ).$ Calculate the flux across C.

For the following exercises, find the flux.

Let $\text{F}=({x}^{2}+{y}^{3})\text{i}+(2xy)\text{j}.$ Calculate flux F orientated counterclockwise across curve C: ${x}^{2}+{y}^{2}=9.$

Find the line integral of ${\int }_{C}^{}{z}^{2}dx+ydy+2ydz,$ where C consists of two parts: ${C}_{1}$ and ${C}_{2}.$ ${C}_{1}$ is the intersection of cylinder ${x}^{2}+{y}^{2}=16$ and plane $z=3$ from (0, 4, 3) to $(-4,0,3).$ ${C}_{2}$ is a line segment from $(-4,0,3)$ to (0, 1, 5).

A spring is made of a thin wire twisted into the shape of a circular helix $x=2\,\cos \,t,\,y=2\,\sin \,t,\,z=t.$ Find the mass of two turns of the spring if the wire has constant mass density.

A thin wire is bent into the shape of a semicircle of radius a. If the linear mass density at point P is directly proportional to its distance from the line through the endpoints, find the mass of the wire.

An object moves in force field $\text{F}(x,y,z)={y}^{2}\text{i}+2(x+1)y\text{j}$ counterclockwise from point (2, 0) along elliptical path ${x}^{2}+4{y}^{2}=4$ to $(-2,0),$ and back to point (2, 0) along the x-axis. How much work is done by the force field on the object?

Find the work done when an object moves in force field $\text{F}(x,y,z)=2x\text{i}-(x+z)\text{j}+(y-x)\text{k}$ along the path given by $\text{r}(t)={t}^{2}\text{i}+({t}^{2}-t)\text{j}+3\text{k},$ $0\le t\le 1.$

If an inverse force field F is given by $\text{F}(x,y,z)=\frac{k}{{\| \text{r}\| }^{3}}\text{r},$ where k is a constant, find the work done by F as its point of application moves along the x-axis from $A(1,0,0)\,\text{to}\,B(2,0,0).$

David and Sandra plan to evaluate line integral ${\int }_{C}^{}\text{F}\cdot d\text{r}$ along a path in the xy-plane from (0, 0) to (1, 1). The force field is $\text{F}(x,y)=(x+2y)\text{i}+(-x+{y}^{2})\text{j}.$ David chooses the path that runs along the x-axis from (0, 0) to (1, 0) and then runs along the vertical line $x=1$ from (1, 0) to the final point (1, 1). Sandra chooses the direct path along the diagonal line $y=x$ from (0, 0) to (1, 1). Whose line integral is larger and by how much?

True or False? If vector field F is conservative on the open and connected region D, then line integrals of F are path independent on D, regardless of the shape of D.

True or False? Function $\text{r}(t)=\text{a}+t(\text{b}-\text{a}),$ where $0\le t\le 1,$ parameterizes the straight-line segment from $\text{a}\,\text{to}\,\text{b}.$