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Equations:

  1. cos ( ϕ + 1 ) = sin ( ϕ + p )
  2. sin ( ϕ ) = - sin ( ϕ + p )
  3. tan ( ϕ ) = sin ( ϕ ) sin ϕ + p
  4. cos ( ϕ ) = - tan ( ϕ + p ) · cos ( ϕ - p )

( tan ( ϕ ) is defined in exercise 1)

A sine wave.

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A cosine wave.

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Trapezoid rule for estimating area (integration)

Basic method

A useful tool for analyzing curves is finding the area underneath them. When we have an unknown combination of waves, we can estimate the area under the curve using the trapezoid rule . We will use f ( t ) = sin ( t ) as an example. If we were to estimate part of the area under the curve with one trapezoid, we might do the following:

A single trapezoid.

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Estimating the area with four trapezoids.

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Applying the trapezoid rule to a sine wave.

We have labeled the two heights, h 1 and h 2 , and the length of the base b . The area of the square is:

area of square = ( base ) × ( height ) = b · h 1

The area of the top triangle is:

area of triangle = ( base ) × ( height ) 2 = b · ( h 2 - h 1 ) 2

The total area of the trapezoid is then:

area of trapezoid = b · h 1 + b · ( h 2 - h 1 ) 2 = b · ( h 2 + h 1 ) 2

If we know that the two points on the x -axis are t 1 and t 2 , then b = t 2 - t 1 . In the figure above, t 1 = . 5 and t 2 = 1 . 5 . Then the heights follow from the function: h 1 = f ( t 1 ) = sin ( t 1 ) and h 2 = f ( t 2 ) = sin ( t 2 ) . Thus in general, the area of a trapezoid approximating the area under f between the points t 1 and t 2 is:

area of trapezoid = b · ( h 2 + h 1 ) 2 = ( t 2 - t 1 ) f ( t 1 ) + f ( t 2 ) 2

In order to get a good estimate, we split up the domain of the function f ( t ) into several intervals [ t i , t i + 1 ] . For each interval, we calculate the area of the trapezoid that approximates the area under that curve. For example, we could approximate f ( t ) over [ 0 , 1 ] using four equal intervals. This would look like:

In this case, our estimate would be

approximate area of curve = ( t 2 - t 1 ) f ( t 1 ) + f ( t 2 ) 2 + + ( t 4 - t 3 ) f ( t 3 ) + f ( t 4 ) 2

As we take smaller and smaller intervals, our approximation will get better, because there will be less space between the trapezoids and the curve. We can prove that the trapezoid rule given `order 2' convergence–that is, if we cut our intervals in half, our error gets four times smaller.

In the general case, if we split up the domain of the function at points { t 1 , t 2 , , t n } , then the rule for the estimate is

approximate area of curve = ( t 1 - t 2 ) f ( t 1 ) + f ( t 2 ) 2 + + ( t n - t n - 1 ) f ( t n - 1 ) + f ( t n ) 2

This formula can be further reduced, which is the subject of Exercise 2.1.

Coding the trapezoid rule

Here we present a code that uses the trapezoid rule to find the area under any function we provide. We need a vector x that holds the values of the domain, for example x = 0:.01:pi . We then need a vector y that holds the function values at those x points, for example y = sin(x) .

function curve_area = mytrapz(x, y, fast) % function curve_area = mytrapz(x, y, fast)% % mytrapz.m performs the trapezoid rule on the vector given by x and y.% Input: %   x - a vector containing the domain of the function%   y - a vector containing values of the function corresponding to the curve_area = 0;%loop through and add up trapezoids for as many points as we are givenfor n = 2 : numel(x)

We start the code with zero area under the curve, since we haven't counted anything yet. Then we create a for loop to count each triangle individually. As we see above, more trapezoids leads to better answers, so we want to use as many trapezoids as we possibly can. In this situation, that means using every point in x and y . The function numel simply counts the number of elements in x . We then calculate the area of the current triangle (within the loop):

Questions & Answers

Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
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Maciej
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Abigail
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Anassong
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
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Devang Reply
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
Abhijith Reply
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?
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.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
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.
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Cied
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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