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  • Determine the length of a curve, y = f ( x ) , between two points.
  • Determine the length of a curve, x = g ( y ) , between two points.
  • Find the surface area of a solid of revolution.

In this section, we use definite integrals to find the arc length of a curve. We can think of arc length    as the distance you would travel if you were walking along the path of the curve. Many real-world applications involve arc length. If a rocket is launched along a parabolic path, we might want to know how far the rocket travels. Or, if a curve on a map represents a road, we might want to know how far we have to drive to reach our destination.

We begin by calculating the arc length of curves defined as functions of x , then we examine the same process for curves defined as functions of y . (The process is identical, with the roles of x and y reversed.) The techniques we use to find arc length can be extended to find the surface area of a surface of revolution, and we close the section with an examination of this concept.

Arc length of the curve y = f ( x )

In previous applications of integration, we required the function f ( x ) to be integrable, or at most continuous. However, for calculating arc length we have a more stringent requirement for f ( x ) . Here, we require f ( x ) to be differentiable, and furthermore we require its derivative, f ( x ) , to be continuous. Functions like this, which have continuous derivatives, are called smooth . (This property comes up again in later chapters.)

Let f ( x ) be a smooth function defined over [ a , b ] . We want to calculate the length of the curve from the point ( a , f ( a ) ) to the point ( b , f ( b ) ) . We start by using line segments to approximate the length of the curve. For i = 0 , 1 , 2 ,… , n , let P = { x i } be a regular partition of [ a , b ] . Then, for i = 1 , 2 ,… , n , construct a line segment from the point ( x i 1 , f ( x i 1 ) ) to the point ( x i , f ( x i ) ) . Although it might seem logical to use either horizontal or vertical line segments, we want our line segments to approximate the curve as closely as possible. [link] depicts this construct for n = 5 .

This figure is a graph in the first quadrant. The curve increases and decreases. It is divided into parts at the points a=xsub0, xsub1, xsub2, xsub3, xsub4, and xsub5=b. Also, there are line segments between the points on the curve.
We can approximate the length of a curve by adding line segments.

To help us find the length of each line segment, we look at the change in vertical distance as well as the change in horizontal distance over each interval. Because we have used a regular partition, the change in horizontal distance over each interval is given by Δ x . The change in vertical distance varies from interval to interval, though, so we use Δ y i = f ( x i ) f ( x i 1 ) to represent the change in vertical distance over the interval [ x i 1 , x i ] , as shown in [link] . Note that some (or all) Δ y i may be negative.

This figure is a graph. It is a curve above the x-axis beginning at the point f(xsubi-1). The curve ends in the first quadrant at the point f(xsubi). Between the two points on the curve is a line segment. A right triangle is formed with this line segment as the hypotenuse, a horizontal segment with length delta x, and a vertical line segment with length delta y.
A representative line segment approximates the curve over the interval [ x i 1 , x i ] .

By the Pythagorean theorem, the length of the line segment is ( Δ x ) 2 + ( Δ y i ) 2 . We can also write this as Δ x 1 + ( ( Δ y i ) / ( Δ x ) ) 2 . Now, by the Mean Value Theorem, there is a point x i * [ x i 1 , x i ] such that f ( x i * ) = ( Δ y i ) / ( Δ x ) . Then the length of the line segment is given by Δ x 1 + [ f ( x i * ) ] 2 . Adding up the lengths of all the line segments, we get

Arc Length i = 1 n 1 + [ f ( x i * ) ] 2 Δ x .

This is a Riemann sum. Taking the limit as n , we have

Questions & Answers

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Damian Reply
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Akash Reply
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Do somebody tell me a best nano engineering book for beginners?
s. Reply
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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.
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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.
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Mostly, they use nano carbon for electronics and for materials to be strengthened.
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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?
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for screen printed electrodes ?
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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
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Damian Reply
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abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
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Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
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silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Leaves accumulate on the forest floor at a rate of 2 g/cm2/yr and also decompose at a rate of 90% per year. Write a differential equation governing the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
Abdul Reply
You have a cup of coffee at temperature 70°C, which you let cool 10 minutes before you pour in the same amount of milk at 1°C as in the preceding problem. How does the temperature compare to the previous cup after 10 minutes?
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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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