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  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

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Eke Reply
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Missy Reply
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Lale Reply
no can't
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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
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..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
has a lot of application modern world
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Jyoti Reply
ya I also want to know the raman spectra
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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.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
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What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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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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