# 6.2 Line integrals  (Page 2/20)

 Page 2 / 20

You may have noticed a difference between this definition of a scalar line integral and a single-variable integral. In this definition, the arc lengths $\text{Δ}{s}_{1},\text{Δ}{s}_{2}\text{,…},\text{Δ}{s}_{n}$ aren’t necessarily the same; in the definition of a single-variable integral, the curve in the x -axis is partitioned into pieces of equal length. This difference does not have any effect in the limit. As we shrink the arc lengths to zero, their values become close enough that any small difference becomes irrelevant.

## Definition

Let $f$ be a function with a domain that includes the smooth curve $C$ that is parameterized by $\text{r}\left(t\right)=⟨x\left(t\right),y\left(t\right),z\left(t\right)⟩,$ $a\le t\le b.$ The scalar line integral    of $f$ along $C$ is

${\int }_{C}f\left(x,y,z\right)ds=\underset{n\to \infty }{\text{lim}}\sum _{i=1}^{n}f\left({P}_{i}^{*}\right)\text{Δ}{s}_{i}$

if this limit exists $\left({t}_{i}^{*}$ and $\text{Δ}{s}_{i}$ are defined as in the previous paragraphs). If C is a planar curve, then C can be represented by the parametric equations $x=x\left(t\right),y=y\left(t\right),$ and $a\le t\le b.$ If C is smooth and $f\left(x,y\right)$ is a function of two variables, then the scalar line integral of $f$ along C is defined similarly as

${\int }_{C}f\left(x,y\right)ds=\underset{n\to \infty }{\text{lim}}\sum _{i=1}^{n}f\left({P}_{i}^{*}\right)\text{Δ}{s}_{i},$

if this limit exists.

If $f$ is a continuous function on a smooth curve C , then ${\int }_{C}fds$ always exists. Since ${\int }_{C}fds$ is defined as a limit of Riemann sums, the continuity of $f$ is enough to guarantee the existence of the limit, just as the integral ${\int }_{a}^{b}g\left(x\right)dx$ exists if g is continuous over $\left[a,b\right].$

Before looking at how to compute a line integral, we need to examine the geometry captured by these integrals. Suppose that $f\left(x,y\right)\ge 0$ for all points $\left(x,y\right)$ on a smooth planar curve $C.$ Imagine taking curve $C$ and projecting it “up” to the surface defined by $f\left(x,y\right),$ thereby creating a new curve ${C}^{\prime }$ that lies in the graph of $f\left(x,y\right)$ ( [link] ). Now we drop a “sheet” from ${C}^{\prime }$ down to the xy-plane. The area of this sheet is ${\int }_{C}f\left(x,y\right)ds.$ If $f\left(x,y\right)\le 0$ for some points in $C,$ then the value of ${\int }_{C}f\left(x,y\right)ds$ is the area above the xy-plane less the area below the xy-plane. (Note the similarity with integrals of the form ${\int }_{a}^{b}g\left(x\right)dx.\right)$

From this geometry, we can see that line integral ${\int }_{C}f\left(x,y\right)ds$ does not depend on the parameterization $\text{r}\left(t\right)$ of C . As long as the curve is traversed exactly once by the parameterization, the area of the sheet formed by the function and the curve is the same. This same kind of geometric argument can be extended to show that the line integral of a three-variable function over a curve in space does not depend on the parameterization of the curve.

## Finding the value of a line integral

Find the value of integral ${\int }_{C}2ds,$ where $C$ is the upper half of the unit circle.

The integrand is $f\left(x,y\right)=2.$ [link] shows the graph of $f\left(x,y\right)=2,$ curve C , and the sheet formed by them. Notice that this sheet has the same area as a rectangle with width $\pi$ and length 2. Therefore, ${\int }_{C}2ds=2\pi .$

To see that ${\int }_{C}2ds=2\pi$ using the definition of line integral, we let $\text{r}\left(t\right)$ be a parameterization of C . Then, $f\left(\text{r}\left({t}_{i}\right)\right)=2$ for any number ${t}_{i}$ in the domain of r . Therefore,

$\begin{array}{cc}\hfill {\int }_{C}fds& =\underset{n\to \infty }{\text{lim}}\sum _{i=1}^{n}f\left(\text{r}\left({t}_{i}^{*}\right)\right)\text{Δ}{s}_{i}\hfill \\ & =\underset{n\to \infty }{\text{lim}}\sum _{i=1}^{n}2\text{Δ}{s}_{i}\hfill \\ & =2\underset{n\to \infty }{\text{lim}}\sum _{i=1}^{n}2\text{Δ}{s}_{i}\hfill \\ & =2\left(\text{length of C}\right)\hfill \\ & =2\pi .\hfill \end{array}$

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
Do somebody tell me a best nano engineering book for beginners?
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
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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!