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  • Find the parametric representations of a cylinder, a cone, and a sphere.
  • Describe the surface integral of a scalar-valued function over a parametric surface.
  • Use a surface integral to calculate the area of a given surface.
  • Explain the meaning of an oriented surface, giving an example.
  • Describe the surface integral of a vector field.
  • Use surface integrals to solve applied problems.

We have seen that a line integral is an integral over a path in a plane or in space. However, if we wish to integrate over a surface (a two-dimensional object) rather than a path (a one-dimensional object) in space, then we need a new kind of integral that can handle integration over objects in higher dimensions. We can extend the concept of a line integral to a surface integral to allow us to perform this integration.

Surface integrals are important for the same reasons that line integrals are important. They have many applications to physics and engineering, and they allow us to develop higher dimensional versions of the Fundamental Theorem of Calculus. In particular, surface integrals allow us to generalize Green’s theorem to higher dimensions, and they appear in some important theorems we discuss in later sections.

Parametric surfaces

A surface integral is similar to a line integral, except the integration is done over a surface rather than a path. In this sense, surface integrals expand on our study of line integrals. Just as with line integrals, there are two kinds of surface integrals: a surface integral of a scalar-valued function and a surface integral of a vector field.

However, before we can integrate over a surface, we need to consider the surface itself. Recall that to calculate a scalar or vector line integral over curve C , we first need to parameterize C . In a similar way, to calculate a surface integral over surface S , we need to parameterize S . That is, we need a working concept of a parameterized surface (or a parametric surface ), in the same way that we already have a concept of a parameterized curve.

A parameterized surface is given by a description of the form

r ( u , v ) = x ( u , v ) , y ( u , v ) , z ( u , v ) .

Notice that this parameterization involves two parameters, u and v , because a surface is two-dimensional, and therefore two variables are needed to trace out the surface. The parameters u and v vary over a region called the parameter domain, or parameter space —the set of points in the uv -plane that can be substituted into r . Each choice of u and v in the parameter domain gives a point on the surface, just as each choice of a parameter t gives a point on a parameterized curve. The entire surface is created by making all possible choices of u and v over the parameter domain.


Given a parameterization of surface r ( u , v ) = x ( u , v ) , y ( u , v ) , z ( u , v ) , the parameter domain of the parameterization is the set of points in the uv -plane that can be substituted into r .

Parameterizing a cylinder

Describe surface S parameterized by

r ( u , v ) = cos u , sin u , v , < u < , < v < .

To get an idea of the shape of the surface, we first plot some points. Since the parameter domain is all of 2 , we can choose any value for u and v and plot the corresponding point. If u = v = 0 , then r ( 0 , 0 ) = 1 , 0 , 0 , so point (1, 0, 0) is on S . Similarly, points r ( π , 2 ) = ( −1 , 0 , 2 ) and r ( π 2 , 4 ) = ( 0 , 1 , 4 ) are on S .

Although plotting points may give us an idea of the shape of the surface, we usually need quite a few points to see the shape. Since it is time-consuming to plot dozens or hundreds of points, we use another strategy. To visualize S , we visualize two families of curves that lie on S. In the first family of curves we hold u constant; in the second family of curves we hold v constant. This allows us to build a “skeleton” of the surface, thereby getting an idea of its shape.

First, suppose that u is a constant K . Then the curve traced out by the parameterization is cos K , sin K , v , which gives a vertical line that goes through point ( cos K , sin K , v ) in the xy -plane.

Now suppose that v is a constant K. Then the curve traced out by the parameterization is cos u , sin u , K , which gives a circle in plane z = K with radius 1 and center (0, 0, K ).

If u is held constant, then we get vertical lines; if v is held constant, then we get circles of radius 1 centered around the vertical line that goes through the origin. Therefore the surface traced out by the parameterization is cylinder x 2 + y 2 = 1 ( [link] ).

Three diagrams in three dimensions. The first shows vertical lines around the origin. The second shows parallel circles all with center at the origin and radius of 1. The third shows the lines and circle. Together, they form the skeleton of a cylinder.
(a) Lines cos K , sin K , v for K = 0 , π 2 , π , and 3 π 2 . (b) Circles cos u , sin u , K for K = −2 , −1 , 1 , and 2 . (c) The lines and circles together. As u and v vary, they describe a cylinder.

Notice that if x = cos u and y = sin u , then x 2 + y 2 = 1 , so points from S do indeed lie on the cylinder. Conversely, each point on the cylinder is contained in some circle cos u , sin u , k for some k , and therefore each point on the cylinder is contained in the parameterized surface ( [link] ).

An image of a vertical cylinder in three dimensions with the center of its circular base located on the z axis.
Cylinder x 2 + y 2 = r 2 has parameterization r ( u , v ) = r cos u , r sin u , v , 0 u 2 π , < v < .
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Questions & Answers

what's a psy.d degree
Robin Reply
Psychology Doctorate. The main difference between PsyD and PhD Psychology is that PsyD mainly focuses on clinical sessions and PhD focuses on research.
so most of the phd holders become professors teaching what they have learnt
Hi, I have a question. What factors influence social facilitation and social inhibition in groups?
Amritpal Reply
it really depends on that person on how the can task in life either they can work better with other people or you do worse working with people.
the task
they task
Do you know of any studies done to show social facilitation and social inhibition
social facilitation is often influenced by the person's perception of the situation and their appraisal of the task. for example, a track athlete who views their competition as a challenge rather than a threat, will tend to run the fastest they have ever ran under a large crowd.
Another condition is if the person has mastered the ability to perform that task. A track athlete who trains everyday, will perform above average under a large crowd, however, someone who does not run, will tend to do worse than their actual ability under a large crowd
social inhibition can be explained through evolutionary biology or attachment personality theory. some people develop an avoidance personality trait, which when they feel under pressure, they tend to leave the group or avoid a gathering due to high lev of stress
Does anyone know how I can transfer my higher national diploma credits in usa?
Qwanta Reply
Oh wow. I never realised there was a chat on this app!
Amritpal Reply
Then welcome dearest
Thank you! ❤️
thank you
Hello Ana, good afternoon from here
Philip Reply
hello im new student here.
Junalyn Reply
o sorry madm
I m also student.
what is easier option to psychology?
I'm new too
As in the the specialisation or ?
good morning
understanding human nature is what we are going to talk about.
okay, go ahead
“ Every child is left to evaluate his experiences for himself, and to take care of his own personal development outside the classroom. There is no tradition for the acquisition of a true knowledge of the human psyche. The science of human nature thus finds itself today in the position that chemis
well, I think every individual is different and to a great extent we can't say confidently that we understand the human nature in its entirety
lack determines what we will become in life?.
Not 100% but it plays a huge role
Hi! I wasn't aware there was a chat feature on this app 😊
👍Philip Anyaegbu
yes Ana I'm here
alternatively we are different, we have basic similar qualities
Yes true Bridget.
hello there
who we are how we think what we do insight
@ bridget. yes that's true
Ana ioanidis? do you speak Greek?
Qwanta no I dont speak Greek my husband does
are you a psychologist? or your husband?
I need some reliable informations about my studies...
not a psychologist
I ask questions
so what is the cause why a person experiences a psychological disorder if it not of their own?
Junalyn Reply
it's partly genetics,culture, environment and substance abuse. generally these are the common factors which can lead to most of the disorders
r u proficcer
u r psychology subject
not a professor but a clinical psychologist
can i ask you, i keep getting headache, and had a history of hypothyroid ans diagnosed as having mild anxiety disorder. i want to finish. my undergraduate psychology thesis.. but always unable to have good ambition and energy to focus on writing the case and the theories... whats the best remedy
had a headache and have a history
297 according to the DSM 5
how many disorders are there
Janareon Reply
interesting question
it is😅
I'm guessing it's in the tens of thousands, maybe hundreds if you include counter interactions and stuff like it. although I'm sure it's almost impossible to know for sure, unless you're very rich and connected to the right people. but as I said: guessing.
There are more than 200 classified forms of mental illness. Some of the more common disorders are: clinical depression, bipolar disorder, dementia, schizophrenia and anxiety disorders. Symptoms may include changes in mood, personality, personal habits and/or social withdrawal. that is what I think
too difficult to number. diagnosing a disorder is just checking off boxes on a compilation of symptoms that might match any particular condition on the DSM
what is psychology of the guest
Lirilong Reply
I don't understand the question can you elaborate?
the study of philosophy gives to the sociologist
Aman Reply
why women are viewed as far more emotional than men?
Wario Reply
because they actually are ,I guess.
May be they are biologically milder than man. It does not mean they are not equal with man. Man can also be emotional and can are oppressed with traditional norms. For example, man are not to be cry, in actual man are also emotional being and they cannot have the right to show their sentiment.
sollungal bro sollungal...
because of she has hormonal fluctuations than men..
usually women are more emotional than men and they are multi talented people
Both men and women are capable of expressing emotions. But women has the highest percentage of doing that. Because our society is conditioned by nature in such a way that men are expected to suppress their emotions and motivate them through their acts or thoughts, which has it's side effects of....
denial of any emotion which they feel that useless at that point. Women in the other hand were encouraged/not controlled to suppress their emotions and let them out what they feel about it. I feel that's the wonderful superpower of the women.
Men's emotion comes mostly with memories triggered by senses. That means, they thoughts have the power to decide whether to let go of emotions or not.
because women tend to be more agreeable than men.
women at a young age are conditioned to be more in tune with emotion than man should be less
The question is "Why women Viewed are as far more emotional than men ?" it's not a question whether women are more emotional than men. This is more an issue about the point of view from the observer, his/her assumption what emotional behavior is or what emotional behavior is.
The anwers are answers more to the question " Are women more emotional than men?"
hi are you on what'sap
Wunuji Reply
no not really a fan of social laziness
is there any way for a Btech E.C.E. graduate to take on MSC. PSYCHOLOGY
ya you can just by cracking entrance , if in India
Any stream in UG can go for M.Sc (Psychology)
classification of traits and how they are measured
chinedu Reply
what was Freud's first name?
Robin Reply
Sigismund Schlomo Freud
Sigmund Freud
Leon Vygotsky and Sergei Rubenstein please tell me contributions of these personalities in in 4 lines
sigmund Freud
Lev Vygotsky was the founder of socio-cultural theory
So psychology was based off of the Greek gods I that right or no?
Tanya Reply
psychology is the study. anything -ology is the study of a certain field. Psyche is a mortal woman who becomes divine in Greek mythology. The etymology of the word "psyche" in Greek means "spirit" or "soul"
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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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