# 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

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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
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