# 6.2 Line integrals  (Page 7/20)

 Page 7 / 20
${\int }_{{a}_{1}}^{{a}_{n}}f\left(x\right)dx={\int }_{{a}_{1}}^{{a}_{2}}f\left(x\right)dx+{\int }_{{a}_{1}}^{{a}_{3}}f\left(x\right)dx+\cdots +{\int }_{{a}_{n-1}}^{{a}_{n}}f\left(x\right)dx,$

which is analogous to property iv.

## Using properties to compute a vector line integral

Find the value of integral ${\int }_{C}\text{F}·\text{T}ds,$ where C is the rectangle (oriented counterclockwise) in a plane with vertices $\left(0,0\right),\left(2,0\right),\left(2,1\right),\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\left(0,1\right),$ and where $\text{F}=⟨x-2y,y-x⟩$ ( [link] ).

Note that curve C is the union of its four sides, and each side is smooth. Therefore C is piecewise smooth. Let ${C}_{1}$ represent the side from $\left(0,0\right)$ to $\left(2,0\right),$ let ${C}_{2}$ represent the side from $\left(2,0\right)$ to $\left(2,1\right),$ let ${C}_{3}$ represent the side from $\left(2,1\right)$ to $\left(0,1\right),$ and let ${C}_{4}$ represent the side from $\left(0,1\right)$ to $\left(0,0\right)$ ( [link] ). Then,

${\int }_{C}\text{F}·\text{T}d\text{r}={\int }_{{C}_{1}}\text{F}·\text{T}d\text{r}+{\int }_{{C}_{2}}\text{F}·\text{T}d\text{r}+{\int }_{{C}_{3}}\text{F}·\text{T}d\text{r}+{\int }_{{C}_{4}}\text{F}·\text{T}d\text{r}.$

We want to compute each of the four integrals on the right-hand side using [link] . Before doing this, we need a parameterization of each side of the rectangle. Here are four parameterizations (note that they traverse C counterclockwise):

$\begin{array}{c}{C}_{1}:⟨t,0⟩,0\le t\le 2\hfill \\ {C}_{2}:⟨2,t⟩,0\le t\le 1\hfill \\ {C}_{3}:⟨2-t,1⟩,0\le t\le 2\hfill \\ {C}_{4}:⟨0,1-t⟩,0\le t\le 1.\hfill \end{array}$

Therefore,

$\begin{array}{cc}\hfill {\int }_{{C}_{1}}\text{F}·\text{T}d\text{r}& ={\int }_{0}^{2}\text{F}\left(\text{r}\left(t\right)\right)·{r}^{\prime }\left(t\right)dt\hfill \\ & ={\int }_{0}^{2}⟨t-2\left(0\right),0-t⟩·⟨1,0⟩dt={\int }_{0}^{1}tdt\hfill \\ & ={\left[\frac{{t}^{2}}{2}\right]}_{0}^{2}=2.\hfill \end{array}$

Notice that the value of this integral is positive, which should not be surprising. As we move along curve C 1 from left to right, our movement flows in the general direction of the vector field itself. At any point along C 1 , the tangent vector to the curve and the corresponding vector in the field form an angle that is less than 90°. Therefore, the tangent vector and the force vector have a positive dot product all along C 1 , and the line integral will have positive value.

The calculations for the three other line integrals are done similarly:

$\begin{array}{cc}\hfill {\int }_{{C}_{2}}\text{F}·d\text{r}& ={\int }_{0}^{1}⟨2-2t,t-2⟩·⟨0,1⟩dt\hfill \\ & ={\int }_{0}^{1}\left(t-2\right)dt\hfill \\ & ={\left[\frac{{t}^{2}}{2}-2t\right]}_{0}^{1}=-\frac{3}{2},\hfill \end{array}$
$\begin{array}{cc}\hfill {\int }_{{C}_{3}}\text{F}·\text{T}ds& ={\int }_{0}^{2}⟨\left(2-t\right)-2,1-\left(2-t\right)⟩·⟨-1,0⟩dt\hfill \\ & ={\int }_{0}^{2}tdt=2,\hfill \end{array}$

and

$\begin{array}{cc}\hfill {\int }_{{C}_{4}}\text{F}·d\text{r}& ={\int }_{0}^{1}⟨-2\left(1-t\right),1-t⟩·⟨0,-1⟩dt\hfill \\ & ={\int }_{0}^{1}\left(t-1\right)dt\hfill \\ & ={\left[\frac{{t}^{2}}{2}-t\right]}_{0}^{1}=-\frac{1}{2}.\hfill \end{array}$

Thus, we have ${\int }_{C}\text{F}·d\text{r}=2.$

Calculate line integral ${\int }_{C}\text{F}·d\text{r},$ where F is vector field $⟨{y}^{2},2xy+1⟩$ and C is a triangle with vertices $\left(0,0\right),$ $\left(4,0\right),$ and $\left(0,5\right),$ oriented counterclockwise.

0

## Applications of line integrals

Scalar line integrals have many applications. They can be used to calculate the length or mass of a wire, the surface area of a sheet of a given height, or the electric potential of a charged wire given a linear charge density. Vector line integrals are extremely useful in physics. They can be used to calculate the work done on a particle as it moves through a force field, or the flow rate of a fluid across a curve. Here, we calculate the mass of a wire using a scalar line integral and the work done by a force using a vector line integral.

Suppose that a piece of wire is modeled by curve C in space. The mass per unit length (the linear density) of the wire is a continuous function $\rho \left(x,y,z\right).$ We can calculate the total mass of the wire using the scalar line integral ${\int }_{C}\rho \left(x,y,z\right)ds.$ The reason is that mass is density multiplied by length, and therefore the density of a small piece of the wire can be approximated by $\rho \left(x*,y*,z*\right)\text{Δ}s$ for some point $\left(x*,y*,z*\right)$ in the piece. Letting the length of the pieces shrink to zero with a limit yields the line integral ${\int }_{C}\rho \left(x,y,z\right)ds.$

## Calculating the mass of a wire

Calculate the mass of a spring in the shape of a curve parameterized by $⟨t,2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t,2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t⟩,$ $0\le t\le \frac{\pi }{2},$ with a density function given by $\rho \left(x,y,z\right)={e}^{x}+yz$ kg/m ( [link] ).

To calculate the mass of the spring, we must find the value of the scalar line integral ${\int }_{C}\left({e}^{x}+yz\right)ds,$ where C is the given helix. To calculate this integral, we write it in terms of t using [link] :

$\begin{array}{cc}\hfill {\int }_{C}{e}^{x}+yzds& ={\int }_{0}^{\pi \text{/}2}\left(\left({e}^{t}+4\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t\right)\sqrt{1+{\left(-2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t\right)}^{2}+{\left(2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t\right)}^{2}}\right)dt\hfill \\ & ={\int }_{0}^{\pi \text{/}2}\left(\left({e}^{t}+4\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t\right)\sqrt{5}\right)dt\hfill \\ & =\sqrt{5}{\left[{e}^{t}+2\phantom{\rule{0.2em}{0ex}}{\text{sin}}^{2}t\right]}_{t=0}^{t=\pi \text{/}2}\hfill \\ & =\sqrt{5\left({e}^{\pi \text{/}2}+1\right)}.\hfill \end{array}$

Therefore, the mass is $\sqrt{5}\left({e}^{\pi \text{/}2}+1\right)$ kg.

how can chip be made from sand
is this allso about nanoscale material
Almas
are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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?