# 6.2 Line integrals  (Page 12/20)

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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?

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?