6.2 Line integrals  (Page 12/20)

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 $(\mathrm{-2},0)?$

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).

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

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

[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 $\text{r}(t)=t\text{i}+t^2\text{j}+\text{ln}t\text{k}\text{,}1\le t\le 3.$

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

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

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

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

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

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

Evaluate ${\displaystyle {\int }_C^{}\text{F}·d\text{r}},$ where $\text{F}(x,y)=2x\text{sin}(y)\text{i}+\left(x^2\text{cos}(y)-3y^2\right)\text{j}$ and

C is any path from $(\mathrm{-1},0)$ to (5, 1).

Find the line integral of $\text{F}(x,y,z)=12x^2\text{i}-5xy\text{j}+xz\text{k}$ 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 ${\displaystyle {\int }_C^{}\left(1+x^{2}y\right)ds},$ where C is ellipse $\text{r}(t)=2\text{cos}t\text{i}+3\text{sin}t\text{j}$ 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 .

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

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

Find the line integral of ${\displaystyle {\int }_C^{}{z}^{2}dx+ydy+2ydz},$ 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 $(\mathrm{-4},0,3).$ $C_2$ is a line segment from $(\mathrm{-4},0,3)$ to (0, 1, 5).

A spring is made of a thin wire twisted into the shape of a circular helix $x=2\text{cos}t\text{,}y=2\text{sin}t\text{,}z=t.$ 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 $x^2+4y^2=4$ to $(\mathrm{-2},0),$ 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 $\text{r}(t)=t^2\text{i}+\left(t^2-t\right)\text{j}+3\text{k},$ $0\le t\le 1.$

If an inverse force field F is given by $\text{F}(x,y,z)=\frac{\text{k}}{{\Vert \text{r}\Vert }^3}\text{r},$ 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 ${\displaystyle {\int }_C^{}\text{F}·d\text{r}}$ along a path in the xy -plane from (0, 0) to (1, 1). The force field is $\text{F}(x,y)=(x+2y)\text{i}+(\text{−}x+y^2)\text{j}.$ 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?