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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 $(\mathrm{-2},0)?$
$W=2\pi $
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).
$\int}_{C}^{}\text{F}\xb7d\text{r}=\frac{25\sqrt{5}+1}{120$
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 $\text{F}(x,y,z)={x}^{2}z\text{i}+6y\text{j}+y{z}^{2}\text{k},$ where C is represented by $\text{r}(t)=t\text{i}+{t}^{2}\text{j}+\text{ln}\phantom{\rule{0.2em}{0ex}}t\text{k}\text{,}\phantom{\rule{0.2em}{0ex}}1\le t\le 3.$
[T] Evaluate line integral ${\int}_{\gamma}^{}x{e}^{y}ds$ where, $\gamma $ is the arc of curve $x={e}^{y}$ from $(1,0)$ to $(e,1).$
${\int}_{\gamma}^{}x{e}^{y}ds\approx 7.157$
[T] Evaluate the integral ${\int}_{\gamma}^{}x{y}^{2}ds},$ where $\gamma $ is a triangle with vertices (0, 1, 2), (1, 0, 3), and $(0,\mathrm{-1},0).$
[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 $(e\text{,}\phantom{\rule{0.2em}{0ex}}1).$
$\int}_{\gamma}^{}\left({y}^{2}-xy\right)dx\approx \mathrm{-1.379$
[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 ${\int}_{C}^{}\text{F}\xb7d\text{r}},$ where $\text{F}(x,y,z)={x}^{2}y\text{i}+(x-z)\text{j}+xyz\text{k}$ and
C : $\text{r}(t)=t\text{i}+{t}^{2}\text{j}+2\text{k}\text{,}\phantom{\rule{0.2em}{0ex}}0\le t\le 1.$
${\int}_{C}^{}\text{F}\xb7d\text{r}}\approx \mathrm{-1.133$
Evaluate ${\int}_{C}^{}\text{F}\xb7d\text{r}},$ where $\text{F}(x,y)=2x\phantom{\rule{0.2em}{0ex}}\text{sin}(y)\text{i}+\left({x}^{2}\text{cos}(y)-3{y}^{2}\right)\text{j}$ and
C is any path from $(\mathrm{-1},0)$ to (5, 1).
Find the line integral of $\text{F}(x,y,z)=12{x}^{2}\text{i}-5xy\text{j}+xz\text{k}$ over path C defined by $y={x}^{2},$ $z={x}^{3}$ from point (0, 0, 0) to point (2, 4, 8).
${\int}_{C}^{}\text{F}\xb7d\text{r}\approx 22.857$
Find the line integral of ${\int}_{C}^{}\left(1+{x}^{2}y\right)ds},$ where C is ellipse $\text{r}(t)=2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{i}+3\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{j}$ 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 .
$\text{flux}=\mathrm{-20}$
Let $\text{F}=\text{\u2212}y\text{i}+x\text{j}$ and let C : $\text{r}(t)=\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{i}+\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{j}$ $(0\le t\le 2\pi ).$ 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 : ${x}^{2}+{y}^{2}=9.$
$\text{flux}=0$
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 $(\mathrm{-4},0,3).$ ${C}_{2}$ 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 $x=2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t\text{,}\phantom{\rule{0.2em}{0ex}}y=2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t\text{,}\phantom{\rule{0.2em}{0ex}}z=t.$ 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 $\text{F}(x,y,z)={y}^{2}\text{i}+2(x+1)y\text{j}$ counterclockwise from point (2, 0) along elliptical path ${x}^{2}+4{y}^{2}=4$ to $(\mathrm{-2},0),$ and back to point (2, 0) along the x -axis. How much work is done by the force field on the object?
$W=0$
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 $A(1,0,0)\phantom{\rule{0.2em}{0ex}}\text{to}\phantom{\rule{0.2em}{0ex}}B(2,0,0).$
$W=\frac{k}{2}$
David and Sandra plan to evaluate line integral $\int}_{C}^{}\text{F}\xb7d\text{r$ along a path in the xy -plane from (0, 0) to (1, 1). The force field is $\text{F}(x,y)=(x+2y)\text{i}+(\text{\u2212}x+{y}^{2})\text{j}.$ 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?
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