# 6.2 Line integrals  (Page 12/20)

 Page 12 / 20

Find the work done by vector field $\text{F}\left(x,y,z\right)=x\text{i}+3xy\text{j}-\left(x+z\right)\text{k}$ 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 $\text{F}\left(x,y\right)=y\text{i}+3x\text{j}$ along the upper part of ellipse $\frac{{x}^{2}}{4}+{y}^{2}=1$ from (2, 0) to $\left(-2,0\right)?$

$W=2\pi$

A vector field is given by $\text{F}\left(x,y\right)=\left(2x+3y\right)\text{i}+\left(3x+2y\right)\text{j}.$ 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).

${\int }_{C}^{}\text{F}·d\text{r}=\frac{25\sqrt{5}+1}{120}$

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

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

${\int }_{C}^{}{y}^{2}dx+\left(xy-{x}^{2}\right)dy=6.15$

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

[T] Evaluate $\text{F}\left(x,y,z\right)={x}^{2}z\text{i}+6y\text{j}+y{z}^{2}\text{k},$ where C is represented by $\text{r}\left(t\right)=t\text{i}+{t}^{2}\text{j}+\text{ln}\phantom{\rule{0.2em}{0ex}}t\text{k}\text{,}\phantom{\rule{0.2em}{0ex}}1\le t\le 3.$

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

${\int }_{\gamma }^{}x{e}^{y}ds\approx 7.157$

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

[T] Evaluate line integral ${\int }_{\gamma }^{}\left({y}^{2}-xy\right)dx,$ where $\gamma$ is curve $y=\text{ln}\phantom{\rule{0.2em}{0ex}}x$ from (1, 0) toward $\left(e\text{,}\phantom{\rule{0.2em}{0ex}}1\right).$

${\int }_{\gamma }^{}\left({y}^{2}-xy\right)dx\approx -1.379$

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

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

C : $\text{r}\left(t\right)=t\text{i}+{t}^{2}\text{j}+2\text{k}\text{,}\phantom{\rule{0.2em}{0ex}}0\le t\le 1.$

${\int }_{C}^{}\text{F}·d\text{r}\approx -1.133$

Evaluate ${\int }_{C}^{}\text{F}·d\text{r},$ where $\text{F}\left(x,y\right)=2x\phantom{\rule{0.2em}{0ex}}\text{sin}\left(y\right)\text{i}+\left({x}^{2}\text{cos}\left(y\right)-3{y}^{2}\right)\text{j}$ and

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

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

${\int }_{C}^{}\text{F}·d\text{r}\approx 22.857$

Find the line integral of ${\int }_{C}^{}\left(1+{x}^{2}y\right)ds,$ where C is ellipse $\text{r}\left(t\right)=2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{i}+3\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{j}$ from $0\le t\le \pi .$

For the following exercises, find the flux.

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

$\text{flux}=-\frac{1}{3}$

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 .

$\text{flux}=-20$

Let $\text{F}=\text{−}y\text{i}+x\text{j}$ and let C : $\text{r}\left(t\right)=\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{i}+\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{j}$ $\left(0\le t\le 2\pi \right).$ Calculate the flux across C .

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

$\text{flux}=0$

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 $\left(-4,0,3\right).$ ${C}_{2}$ is a line segment from $\left(-4,0,3\right)$ to (0, 1, 5).

A spring is made of a thin wire twisted into the shape of a circular helix $x=2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{,}\phantom{\rule{0.2em}{0ex}}y=2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{,}\phantom{\rule{0.2em}{0ex}}z=t.$ Find the mass of two turns of the spring if the wire has constant mass density.

$m=4\pi \rho \sqrt{5}$

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

$W=0$

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

If an inverse force field F is given by $\text{F}\left(x,y,z\right)=\frac{\text{k}}{{‖\text{r}‖}^{3}}\text{r},$ where k is a constant, find the work done by F as its point of application moves along the x -axis from $A\left(1,0,0\right)\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}B\left(2,0,0\right).$

$W=\frac{k}{2}$

David and Sandra plan to evaluate line integral ${\int }_{C}^{}\text{F}·d\text{r}$ along a path in the xy -plane from (0, 0) to (1, 1). The force field is $\text{F}\left(x,y\right)=\left(x+2y\right)\text{i}+\left(\text{−}x+{y}^{2}\right)\text{j}.$ 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?

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