<< Chapter < Page Chapter >> Page >

In addition to parameterizing surfaces given by equations or standard geometric shapes such as cones and spheres, we can also parameterize surfaces of revolution. Therefore, we can calculate the surface area of a surface of revolution by using the same techniques. Let y = f ( x ) 0 be a positive single-variable function on the domain a x b and let S be the surface obtained by rotating f about the x -axis ( [link] ). Let θ be the angle of rotation. Then, S can be parameterized with parameters x and θ by

r ( x , θ ) = x , f ( x ) cos θ , f ( x ) sin θ , a x b , 0 x < 2 π .
Two diagrams, a and b, showing the surface of revolution. The first shows three dimensions. In the (x,y) plane, a curve labeled y = f(x) is drawn in quadrant 1. A line is drawn from the endpoint of the curve down to the x axis, and it is labeled f(x). The second shows the same three dimensional view. However, the curve from the first diagram has been rotated to form a three dimensional shape about the x axis. The boundary is still labeled y = f(x), as the curve in the first plane was. The opening of the three dimensional shape is circular with the radius f(x), just as the line from the curve to the x axis in the plane of the first diagram was labeled. A point on the opening’s boundary is labeled (x,y,z), the distance from the x axis to this point is drawn and labeled f(x), and the height is drawn and labeled z. The height is perpendicular to the x,y plane and, as such, the original f(x) line drawn from the first diagram. The angle between this line and the line from the x axis to (x,y,z) is labeled theta.
We can parameterize a surface of revolution by r ( x , θ ) = x , f ( x ) cos θ , f ( x ) sin θ , a x b , 0 x < 2 π .

Calculating surface area

Find the area of the surface of revolution obtained by rotating y = x 2 , 0 x b about the x -axis ( [link] ).

A solid of revolution drawn in two dimensions. The solid is formed by rotating the function y = x^2 about the x axis. A point C is marked on the x axis between 0 and x’, which marks the opening of the solid.
A surface integral can be used to calculate the surface area of this solid of revolution.

This surface has parameterization

r ( x , θ ) = x , x 2 cos θ , x 2 sin θ , 0 x b , 0 x < 2 π .

The tangent vectors are t x = 1 , 2 x cos θ , 2 x sin θ and t θ = 0 , x 2 sin θ , x 2 cos θ . Therefore,

t x × t θ = 2 x 3 cos 2 θ + 2 x 3 sin 2 θ , x 2 cos θ , x 2 sin θ = 2 x 3 , x 2 cos θ , x 2 sin θ


t x × t θ = 4 x 6 + x 4 cos 2 θ + x 4 sin 2 θ = 4 x 6 + x 4 = x 2 4 x 2 + 1 .

The area of the surface of revolution is

0 b 0 π x 2 4 x 2 + 1 d θ d x = 2 π 0 b x 2 4 x 2 + 1 d x = 2 π [ 1 64 ( 2 4 x 2 + 1 ( 8 x 3 + x ) sinh −1 ( 2 x ) ) ] 0 b = 2 π [ 1 64 ( 2 4 b 2 + 1 ( 8 b 3 + b ) sinh −1 ( 2 b ) ) ] .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Use [link] to find the area of the surface of revolution obtained by rotating curve y = sin x , 0 x π about the x -axis.

2 π ( 2 + sinh −1 ( 1 ) )

Got questions? Get instant answers now!

Surface integral of a scalar-valued function

Now that we can parameterize surfaces and we can calculate their surface areas, we are able to define surface integrals. First, let’s look at the surface integral of a scalar-valued function. Informally, the surface integral of a scalar-valued function is an analog of a scalar line integral in one higher dimension. The domain of integration of a scalar line integral is a parameterized curve (a one-dimensional object); the domain of integration of a scalar surface integral is a parameterized surface (a two-dimensional object). Therefore, the definition of a surface integral follows the definition of a line integral quite closely. For scalar line integrals, we chopped the domain curve into tiny pieces, chose a point in each piece, computed the function at that point, and took a limit of the corresponding Riemann sum. For scalar surface integrals, we chop the domain region (no longer a curve) into tiny pieces and proceed in the same fashion.

Let S be a piecewise smooth surface with parameterization r ( u , v ) = x ( u , v ) , y ( u , v ) , z ( u , v ) with parameter domain D and let f ( x , y , z ) be a function with a domain that contains S. For now, assume the parameter domain D is a rectangle, but we can extend the basic logic of how we proceed to any parameter domain (the choice of a rectangle is simply to make the notation more manageable). Divide rectangle D into subrectangles D i j with horizontal width Δ u and vertical length Δ v . Suppose that i ranges from 1 to m and j ranges from 1 to n so that D is subdivided into mn rectangles. This division of D into subrectangles gives a corresponding division of S into pieces S i j . Choose point P i j in each piece S i j , evaluate P i j at f , and multiply by area Δ S i j to form the Riemann sum

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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
Privacy Information Security Software Version 1.1a
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?