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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

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?
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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
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what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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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.
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
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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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