선적분

In the following exercises, find the work done by force field F on an object moving along the indicated path.

$\text{F}(x,y)=-x\text{i}-2y\text{j}$

$C:\,y={x}^{3}\,\text{from (0, 0) to (2, 8)}$

In the following exercises, find the work done by force field F on an object moving along the indicated path.

$\text{F}(x,\,y)=2xi+y\text{j}$

C: counterclockwise around the triangle with vertices (0, 0), (1, 0), and (1, 1)

In the following exercises, find the work done by force field F on an object moving along the indicated path.

$\text{F}(x,\,y,\,z)=x\text{i}+y\text{j}-5z\text{k}$

$\text{C}:\,\text{r}(t)=2\,\cos \,t\text{i}+2\,\sin \,t\text{j}+t\text{k},\,0\le t\le 2\pi$

Let F be vector field $\text{F}(x,y)=({y}^{2}+2x{e}^{y}+1)\text{i}+(2xy+{x}^{2}{e}^{y}+2y)\text{j}.$ Compute the work of integral ${\int }_{C}^{}\text{F}\cdot d\text{r},$ where C is the path $\text{r}(t)=\sin \,t\text{i}+\cos \,t\text{j},\,0\le t\le \frac{\pi }{2}.$

Compute the work done by force $\text{F}(x,y,z)=2x\text{i}+3y\text{j}-z\text{k}$ along path $\text{r}(t)=t\text{i}+{t}^{2}\text{j}+{t}^{3}\text{k},$ where $0\le t\le 1.$

Evaluate ${\int }_{C}^{}\text{F}\cdot d\text{r},$ where $\text{F}(x,y)=\frac{1}{x+y}\text{i}+\frac{1}{x+y}\text{j}$ and C is the segment of the unit circle going counterclockwise from $(1,0)$ to (0, 1).

Find the work done by vector field $\text{F}(x,y,z)=x\text{i}+3xy\text{j}-(x+z)\text{k}$ on a particle moving along a line segment that goes from (1, 4, 2) to (0, 5, 1).

How much work is required to move an object in vector field $\text{F}(x,y)=y\text{i}+3x\text{j}$ along the upper part of ellipse $\frac{{x}^{2}}{4}+{y}^{2}=1$ from (2, 0) to $(-2,0)?$

A vector field is given by $\text{F}(x,y)=(2x+3y)\text{i}+(3x+2y)\text{j}.$ Evaluate the line integral of the field around a circle of unit radius traversed in a clockwise fashion.

Evaluate the line integral of scalar function $xy$ along parabolic path $y={x}^{2}$ connecting the origin to point (1, 1).

In the following exercises, find the work done by force field F on an object moving along the indicated path.

Find ${\int }_{C}^{}{y}^{2}dx+(xy-{x}^{2})dy$ along C: $y=3x$ from (0, 0) to (1, 3).

In the following exercises, find the work done by force field F on an object moving along the indicated path.

Find ${\int }_{C}^{}{y}^{2}dx+(xy-{x}^{2})dy$ along C: ${y}^{2}=9x$ from (0, 0) to (1, 3).

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

[T] Evaluate $\text{F}(x,y,z)={x}^{2}z\text{i}+6y\text{j}+y{z}^{2}\text{k},$ where C is represented by $\text{r}(t)=t\text{i}+{t}^{2}\text{j}+\ln \,t\text{k},\,1\le t\le 3.$

[T] Evaluate line integral ${\int }_{\gamma }^{}x{e}^{y}ds$ where, $\gamma$ is the arc of curve $x={e}^{y}$ from $(1,0)$ to $(e,1).$

[T] Evaluate the integral ${\int }_{\gamma }^{}x{y}^{2}ds,$ where $\gamma$ is a triangle with vertices (0, 1, 2), (1, 0, 3), and $(0,-1,0).$

[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).