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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

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Good
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?
Radek Reply
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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