# 2.4 Arc length of a curve and surface area

 Page 1 / 8
• Determine the length of a curve, $y=f\left(x\right),$ between two points.
• Determine the length of a curve, $x=g\left(y\right),$ 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 $f\left(x\right)$ to be integrable, or at most continuous. However, for calculating arc length we have a more stringent requirement for $f\left(x\right).$ Here, we require $f\left(x\right)$ to be differentiable, and furthermore we require its derivative, ${f}^{\prime }\left(x\right),$ to be continuous. Functions like this, which have continuous derivatives, are called smooth . (This property comes up again in later chapters.)

Let $f\left(x\right)$ be a smooth function defined over $\left[a,b\right].$ We want to calculate the length of the curve from the point $\left(a,f\left(a\right)\right)$ to the point $\left(b,f\left(b\right)\right).$ We start by using line segments to approximate the length of the curve. For $i=0,\phantom{\rule{0.2em}{0ex}}1,2\text{,…},n,$ let $P=\left\{{x}_{i}\right\}$ be a regular partition of $\left[a,b\right].$ Then, for $i=1,2\text{,…},n,$ construct a line segment from the point $\left({x}_{i-1},f\left({x}_{i-1}\right)\right)$ to the point $\left({x}_{i},f\left({x}_{i}\right)\right).$ 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 $n=5.$

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 $\text{Δ}x.$ The change in vertical distance varies from interval to interval, though, so we use $\text{Δ}{y}_{i}=f\left({x}_{i}\right)-f\left({x}_{i-1}\right)$ to represent the change in vertical distance over the interval $\left[{x}_{i-1},{x}_{i}\right],$ as shown in [link] . Note that some (or all) $\text{Δ}{y}_{i}$ may be negative.

By the Pythagorean theorem, the length of the line segment is $\sqrt{{\left(\text{Δ}x\right)}^{2}+{\left(\text{Δ}{y}_{i}\right)}^{2}}.$ We can also write this as $\text{Δ}x\sqrt{1+{\left(\left(\text{Δ}{y}_{i}\right)\text{/}\left(\text{Δ}x\right)\right)}^{2}}.$ Now, by the Mean Value Theorem, there is a point ${x}_{i}^{*}\in \left[{x}_{i-1},{x}_{i}\right]$ such that ${f}^{\prime }\left({x}_{i}^{*}\right)=\left(\text{Δ}{y}_{i}\right)\text{/}\left(\text{Δ}x\right).$ Then the length of the line segment is given by $\text{Δ}x\sqrt{1+{\left[{f}^{\prime }\left({x}_{i}^{*}\right)\right]}^{2}}.$ Adding up the lengths of all the line segments, we get

$\text{Arc Length}\phantom{\rule{0.2em}{0ex}}\approx \sum _{i=1}^{n}\sqrt{1+{\left[{f}^{\prime }\left({x}_{i}^{*}\right)\right]}^{2}}\phantom{\rule{0.2em}{0ex}}\text{Δ}x.$

This is a Riemann sum. Taking the limit as $n\to \infty ,$ we have

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
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?
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
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?
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?
Abdul