<< Chapter < Page Chapter >> Page >

Find the value of C ( x = y ) d s , where C is the curve parameterized by x = t , y = t , 0 t 1 .

2

Got questions? Get instant answers now!

Note that in a scalar line integral, the integration is done with respect to arc length s , which can make a scalar line integral difficult to calculate. To make the calculations easier, we can translate C f d s to an integral with a variable of integration that is t .

Let r ( t ) = x ( t ) , y ( t ) , z ( t ) for a t b be a parameterization of C . Since we are assuming that C is smooth, r ( t ) = x ( t ) , y ( t ) , z ( t ) is continuous for all t in [ a , b ] . In particular, x ( t ) , y ( t ) , and z ( t ) exist for all t in [ a , b ] . According to the arc length formula, we have

length ( C i ) = Δ s i = t i 1 t i r ( t ) d t .

If width Δ t i = t i t i 1 is small, then function t i 1 t i r ( t ) d t r ( t i * ) Δ t i , r ( t ) is almost constant over the interval [ t i 1 , t i ] . Therefore,

t i 1 t i r ( t ) d t r ( t i * ) Δ t i ,

and we have

i = 1 n f ( r ( t i * ) ) Δ s i = i = 1 n f ( r ( t i * ) ) r ( t i * ) Δ t i .

See [link] .

A segment of an increasing concave down curve labeled C. A small segment of the curved is boxed and labeled as delta t_i. In the zoomed-in insert, this boxed segment of the curve is almost linear.
If we zoom in on the curve enough by making Δ t i very small, then the corresponding piece of the curve is approximately linear.

Note that

lim n i = 1 n f ( r ( t i * ) ) r ( t i * ) Δ t i = a b f ( r ( t ) ) r ( t ) d t .

In other words, as the widths of intervals [ t i 1 , t i ] shrink to zero, the sum i = 1 n f ( r ( t i * ) ) r ( t i * ) Δ t i converges to the integral a b f ( r ( t ) ) r ( t ) d t . Therefore, we have the following theorem.

Evaluating a scalar line integral

Let f be a continuous function with a domain that includes the smooth curve C with parameterization r ( t ) , a t b . Then

C f d s = a b f ( r ( t ) ) r ( t ) d t .

Although we have labeled [link] as an equation, it is more accurately considered an approximation because we can show that the left-hand side of [link] approaches the right-hand side as n . In other words, letting the widths of the pieces shrink to zero makes the right-hand sum arbitrarily close to the left-hand sum. Since

r ( t ) = ( x ( t ) ) 2 + ( y ( t ) ) 2 + ( z ( t ) ) 2 ,

we obtain the following theorem, which we use to compute scalar line integrals.

Scalar line integral calculation

Let f be a continuous function with a domain that includes the smooth curve C with parameterization r ( t ) = x ( t ) , y ( t ) , z ( t ) , a t b . Then

C f ( x , y , z ) d s = a b f ( r ( t ) ) = ( x ( t ) ) 2 + ( y ( t ) ) 2 + ( z ( t ) ) 2 d t .

Similarly,

C f ( x , y ) d s = a b f ( r ( t ) ) = ( x ( t ) ) 2 + ( y ( t ) ) 2 d t

if C is a planar curve and f is a function of two variables.

Note that a consequence of this theorem is the equation d s = r ( t ) d t . In other words, the change in arc length can be viewed as a change in the t domain, scaled by the magnitude of vector r ( t ) .

Evaluating a line integral

Find the value of integral C ( x 2 + y 2 + z ) d s , where C is part of the helix parameterized by r ( t ) = cos t , sin t , t , 0 t 2 π .

To compute a scalar line integral, we start by converting the variable of integration from arc length s to t . Then, we can use [link] to compute the integral with respect to t . Note that f ( r ( t ) ) = cos 2 t + sin 2 t + t = 1 + t and

( x ( t ) ) 2 + ( y ( t ) ) 2 + ( z ( t ) ) 2 = ( sin ( t ) ) 2 + cos 2 ( t ) + 1 = 2 .

Therefore,

C ( x 2 + y 2 + z ) d s = 0 2 π ( 1 + t ) 2 d t .

Notice that [link] translated the original difficult line integral into a manageable single-variable integral. Since

0 2 π ( 1 + t ) 2 d t = [ 2 t + 2 t 2 2 ] 0 2 π = 2 2 π + 2 2 π 2 ,

we have

C ( x 2 + y 2 + z ) d s = 2 2 π + 2 2 π 2 .
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Evaluate C ( x 2 + y 2 + z ) d s , where C is the curve with parameterization r ( t ) = sin ( 3 t ) , cos ( 3 t ) , 0 t π 4 .

1 3 + 2 6 + 3 π 4

Got questions? Get instant answers now!

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
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?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 8

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Calculus volume 3' conversation and receive update notifications?

Ask