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Key equations

  • Scalar surface integral
    S f ( x , y , z ) d S = D f ( r ( u , v ) ) | | t u × t v | | d A
  • Flux integral
    S F · N d S = S F · d S = D F ( r ( u , v ) ) · ( t u × t v ) d A

For the following exercises, determine whether the statements are true or false .

If surface S is given by { ( x , y , z ) : 0 x 1 , 0 y 1 , z = 10 } , then S f ( x , y , z ) d S = 0 1 0 1 f ( x , y , 10 ) d x d y .

True

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If surface S is given by { ( x , y , z ) : 0 x 1 , 0 y 1 , z = x } , then S f ( x , y , z ) d S = 0 1 0 1 f ( x , y , x ) d x d y .

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Surface r = v cos u , v sin u , v 2 , for 0 u π , 0 v 2 , is the same as surface r = v cos 2 u , v sin 2 u , v , for 0 u π 2 , 0 v 4 .

True

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Given the standard parameterization of a sphere, normal vectors t u × t v are outward normal vectors.

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For the following exercises, find parametric descriptions for the following surfaces.

Plane 3 x 2 y + z = 2

r ( u , v ) = u , v , 2 3 u + 2 v for u < and v < .

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Paraboloid z = x 2 + y 2 , for 0 z 9 .

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Plane 2 x 4 y + 3 z = 16

r ( u , v ) = u , v , 1 3 ( 16 2 u + 4 v ) for | u | < and | v | < .

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The frustum of cone z 2 = x 2 + y 2 , for 2 z 8

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The portion of cylinder x 2 + y 2 = 9 in the first octant, for 0 z 3

A diagram in three dimensions of a section of a cylinder with radius 3. The center of its circular top is (0,0,3). The section exists for x, y, and z between 0 and 3.

r ( u , v ) = 3 cos u , 3 sin u , v for 0 u π 2 , 0 v 3

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A cone with base radius r and height h , where r and h are positive constants

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For the following exercises, use a computer algebra system to approximate the area of the following surfaces using a parametric description of the surface.

[T] Half cylinder { ( r , θ , z ) : r = 4 , 0 θ π , 0 z 7 }

A = 87.9646

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[T] Plane z = 10 x y above square | x | 2 , | y | 2

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For the following exercises, let S be the hemisphere x 2 + y 2 + z 2 = 4 , with z 0 , and evaluate each surface integral, in the counterclockwise direction.

S z d S

S z d S = 8 π

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S ( x 2 + y 2 ) z d S

S ( x 2 + y 2 ) z d S = 16 π

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For the following exercises, evaluate S F · N d s for vector field F , where N is an outward normal vector to surface S.

F ( x , y , z ) = x i + 2 y j = 3 z k , and S is that part of plane 15 x 12 y + 3 z = 6 that lies above unit square 0 x 1 , 0 y 1 .

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F ( x , y ) = x i + y j , and S is hemisphere z = 1 x 2 y 2 .

S F · N d S = 4 π 3

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F ( x , y , z ) = x 2 i + y 2 j + z 2 k , and S is the portion of plane z = y + 1 that lies inside cylinder x 2 + y 2 = 1 .

A cylinder and an intersecting plane shown in three-dimensions. S is the portion of the plane z = y + 1 inside the cylinder x^2 + y ^2 = 1.
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For the following exercises, approximate the mass of the homogeneous lamina that has the shape of given surface S. Round to four decimal places.

[T] S is surface z = 4 x 2 y , with z 0 , x 0 , y 0 ; ξ = x .

m 13.0639

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[T] S is surface z = x 2 + y 2 , with z 1 ; ξ = z .

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[T] S is surface x 2 + y 2 + x 2 = 5 , with z 1 ; ξ = θ 2 .

m 228.5313

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Evaluate S ( y 2 z i + y 3 j + x z k ) · d S , where S is the surface of cube −1 x 1 , −1 y 1 , and 0 z 2 . in a counterclockwise direction.

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Evaluate surface integral S g d S , where g ( x , y , z ) = x z + 2 x 2 3 x y and S is the portion of plane 2 x 3 y + z = 6 that lies over unit square R : 0 x 1 , 0 y 1 .

S g d S = 3 4

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Evaluate S ( x + y + z ) d S , where S is the surface defined parametrically by R ( u , v ) = ( 2 u + v ) i + ( u 2 v ) j + ( u + 3 v ) k for 0 u 1 , and 0 v 2 .

A three-dimensional diagram of the given surface, which appears to be a steeply sloped plane stretching through the (x,y) plane.
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[T] Evaluate S ( x y 2 + z ) d S , where S is the surface defined by R ( u , v ) = u 2 i + v j + u k , 0 u 1 , 0 v 1 .

A three-dimensional diagram of the given surface, which appears to be a curve with edges parallel to the y-axis. It increases in x components and decreases in z components the further it is from the y axis.

S ( x 2 + y z ) d S 0.9617

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[T] Evaluate where S is the surface defined by R ( u , v ) = u i u 2 j + v k , 0 u 2 , 0 v 1 . for 0 u 1 , 0 v 2 .

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Evaluate S ( x 2 + y 2 ) d S , where S is the surface bounded above hemisphere z = 1 x 2 y 2 , and below by plane z = 0 .

S ( x 2 + y 2 ) d S = 4 π 3

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Evaluate S ( x 2 + y 2 + z 2 ) d S , where S is the portion of plane z = x + 1 that lies inside cylinder x 2 + y 2 = 1 .

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[T] Evaluate S x 2 z d S , where S is the portion of cone z 2 = x 2 + y 2 that lies between planes z = 1 and z = 4 .

A diagram of the given upward opening cone in three dimensions. The cone is cut by planes z=1 and z=4.

S x 2 z d S = 1023 2 π 5

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[T] Evaluate S ( x z / y ) d S , where S is the portion of cylinder x = y 2 that lies in the first octant between planes z = 0 , z = 5 , y = 1 , and y = 4 .

A diagram of the given cylinder in three-dimensions. It is cut by the planes z=0, z=5, y=1, and y=4.
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Questions & Answers

Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
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

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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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