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True or False? If C is given by x ( t ) = t , y ( t ) = t , 0 t 1 , then C x y d s = 0 1 t 2 d t .

False

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For the following exercises, use a computer algebra system (CAS) to evaluate the line integrals over the indicated path.

[T] C ( x + y ) d s

C : x = t , y = ( 1 t ) , z = 0 from (0, 1, 0) to (1, 0, 0)

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[T] C ( x y ) d s

C : r ( t ) = 4 t i + 3 t j when 0 t 2

C ( x y ) d s = 10

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[T] C ( x 2 + y 2 + z 2 ) d s

C : r ( t ) = sin t i + cos t j + 8 t k when 0 t π 2

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[T] Evaluate C x y 4 d s , where C is the right half of circle x 2 + y 2 = 16 and is traversed in the clockwise direction.

C x y 4 d s = 8192 5

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[T] Evaluate C 4 x 3 d s , where C is the line segment from ( −2 , −1 ) to (1, 2).

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For the following exercises, find the work done.

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

W = 8

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Find the work done by a person weighing 150 lb walking exactly one revolution up a circular, spiral staircase of radius 3 ft if the person rises 10 ft.

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Find the work done by force field F ( x , y , z ) = 1 2 x i 1 2 y j + 1 4 k on a particle as it moves along the helix r ( t ) = cos t i + sin t j + t k from point ( 1 , 0 , 0 ) to point ( −1 , 0 , 3 π ) .

W = 3 π 4

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Find the work done by vector field F ( x , y ) = y i + 2 x j in moving an object along path C , which joins points (1, 0) and (0, 1).

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Find the work done by force F ( x , y ) = 2 y i + 3 x j + ( x + y ) k in moving an object along curve r ( t ) = cos ( t ) i + sin ( t ) j + 1 6 k , where 0 t 2 π .

W = π

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Find the mass of a wire in the shape of a circle of radius 2 centered at (3, 4) with linear mass density ρ ( x , y ) = y 2 .

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For the following exercises, evaluate the line integrals.

Evaluate C F · d r , where F ( x , y ) = −1 j , and C is the part of the graph of y = 1 2 x 3 x from ( 2 , 2 ) to ( −2 , −2 ) .

C F · d r = 4

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Evaluate γ ( x 2 + y 2 + z 2 ) −1 d s , where γ is the helix x = cos t , y = sin t , z = t ( 0 t T ) .

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Evaluate C y z d x + x z d y + x y d z over the line segment from ( 1 , 1 , 1 ) to ( 3 , 2 , 0 ) .

C y z d x + x z d y + x y d z = −1

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Let C be the line segment from point (0, 1, 1) to point (2, 2, 3). Evaluate line integral C y d s .

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[T] Use a computer algebra system to evaluate the line integral C y 2 d x + x d y , where C is the arc of the parabola x = 4 y 2 from (−5, −3) to (0, 2).

C ( y 2 ) d x + ( x ) d y = 245 6

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[T] Use a computer algebra system to evaluate the line integral C ( x + 3 y 2 ) d y over the path C given by x = 2 t , y = 10 t , where 0 t 1 .

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[T] Use a CAS to evaluate line integral C x y d x + y d y over path C given by x = 2 t , y = 10 t , where 0 t 1 .

C x y d x + y d y = 190 3

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Evaluate line integral C ( 2 x y ) d x + ( x + 3 y ) d y , where C lies along the x -axis from x = 0 to x = 5 .

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[T] Use a CAS to evaluate C y 2 x 2 y 2 d s , where C is x = t , y = t , 1 t 5 .

C y 2 x 2 y 2 d s = 2 ln 5

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[T] Use a CAS to evaluate C x y d s , where C is x = t 2 , y = 4 t , 0 t 1 .

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

F ( x , y ) = x i 2 y j

C : y = x 3 from (0, 0) to (2, 8)

W = −66

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F ( x , y ) = 2 x i + y j

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

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F ( x , y , z ) = x i + y j 5 z k

C : r ( t ) = 2 cos t i + 2 sin t j + t k , 0 t 2 π

W = −10 π 2

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Let F be vector field F ( x , y ) = ( y 2 + 2 x e y + 1 ) i + ( 2 x y + x 2 e y + 2 y ) j . Compute the work of integral C F · d r , where C is the path r ( t ) = sin t i + cos t j , 0 t π 2 .

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Compute the work done by force F ( x , y , z ) = 2 x i + 3 y j z k along path r ( t ) = t i + t 2 j + t 3 k , where 0 t 1 .

W = 2

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Evaluate C F · d r , where F ( x , y ) = 1 x + y i + 1 x + y j and C is the segment of the unit circle going counterclockwise from ( 1 , 0 ) to (0, 1).

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Force F ( x , y , z ) = z y i + x j + z 2 x k acts on a particle that travels from the origin to point (1, 2, 3). Calculate the work done if the particle travels:

  1. along the path ( 0 , 0 , 0 ) ( 1 , 0 , 0 ) ( 1 , 2 , 0 ) ( 1 , 2 , 3 ) along straight-line segments joining each pair of endpoints;
  2. along the straight line joining the initial and final points.
  3. Is the work the same along the two paths?
    A curve and vector field in three dimensions. The curve segments go from (1,2,3) to (0,0,0) to (1,0,0) to (1,2,0), and the arrowheads point to (0,0,0), (1,0,0), and (1,2,0). The surrounding vectors are larger the more the z component increases.

a. W = 11 ; b. W = 11 ; c. Yes

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Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Practice Key Terms 8

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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