# 6.6 Surface integrals  (Page 6/27)

 Page 6 / 27

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 $y=f\left(x\right)\ge 0$ be a positive single-variable function on the domain $a\le x\le b$ and let S be the surface obtained by rotating $f$ about the x -axis ( [link] ). Let $\theta$ be the angle of rotation. Then, S can be parameterized with parameters x and $\theta$ by

$\text{r}\left(x,\theta \right)=⟨x,f\left(x\right)\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,f\left(x\right)\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ⟩,a\le x\le b,0\le x<2\pi .$

## Calculating surface area

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

This surface has parameterization

$\text{r}\left(x,\theta \right)=⟨x,{x}^{2}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,{x}^{2}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ⟩,0\le x\le b,0\le x<2\pi .$

The tangent vectors are ${\text{t}}_{x}=⟨1,2x\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,2x\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ⟩\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}{\text{t}}_{\theta }=⟨0,\text{−}{x}^{2}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ,\text{−}{x}^{2}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ⟩.$ Therefore,

$\begin{array}{cc}\hfill {\text{t}}_{x}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{\text{t}}_{\theta }& =⟨2{x}^{3}{\text{cos}}^{2}\theta +2{x}^{3}{\text{sin}}^{2}\theta ,\text{−}{x}^{2}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,\text{−}{x}^{2}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ⟩\hfill \\ & =⟨2{x}^{3},\text{−}{x}^{2}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta ,\text{−}{x}^{2}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta ⟩\hfill \end{array}$

and

$\begin{array}{cc}\hfill {\text{t}}_{x}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{\text{t}}_{\theta }& =\sqrt{4{x}^{6}+{x}^{4}{\text{cos}}^{2}\theta +{x}^{4}{\text{sin}}^{2}\theta }\hfill \\ & =\sqrt{4{x}^{6}+{x}^{4}}\hfill \\ & ={x}^{2}\sqrt{4{x}^{2}+1}.\hfill \end{array}$

The area of the surface of revolution is

$\begin{array}{cc}\hfill {\int }_{0}^{b}{\int }_{0}^{\pi }{x}^{2}\sqrt{4{x}^{2}+1}d\theta dx& =2\pi {\int }_{0}^{b}{x}^{2}\sqrt{4{x}^{2}+1}dx\hfill \\ & =2\pi {\left[\frac{1}{64}\left(2\sqrt{4{x}^{2}+1}\left(8{x}^{3}+x\right){\text{sinh}}^{-1}\left(2x\right)\right)\right]}_{0}^{b}\hfill \\ & =2\pi \left[\frac{1}{64}\left(2\sqrt{4{b}^{2}+1}\left(8{b}^{3}+b\right){\text{sinh}}^{-1}\left(2b\right)\right)\right].\hfill \end{array}$

Use [link] to find the area of the surface of revolution obtained by rotating curve $y=\text{sin}\phantom{\rule{0.2em}{0ex}}x,0\le x\le \pi$ about the x -axis.

$2\pi \left(\sqrt{2}+{\text{sinh}}^{-1}\left(1\right)\right)$

## 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 $\text{r}\left(u,v\right)=⟨x\left(u,v\right),y\left(u,v\right),z\left(u,v\right)⟩$ with parameter domain D and let $f\left(x,y,z\right)$ 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 ${D}_{ij}$ with horizontal width $\text{Δ}u$ and vertical length $\text{Δ}v.$ 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 ${S}_{ij}.$ Choose point ${P}_{ij}$ in each piece ${S}_{ij},$ evaluate ${P}_{ij}$ at $f$ , and multiply by area $\text{Δ}{S}_{ij}$ to form the Riemann sum

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
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
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
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?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!