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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

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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit 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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