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#1 2016-09-05 19:10:15

peter010
Guest

Integration as a series summaion

Hello,

Integration can be defined as:

b               b
∫ f(x).dx = ∑ f(x)
a               a

Now, If I want to find the integration for f(x)=10, where a=1, b=5 .. I ll got two different results!!!

(1)

5
∫ 10 = (10*5) - (10*1) = 40
1               

(2)

5
∑ 10 = 10 + 10 + 10 +10 +10 = 50, or: = (10 *5) = 50

-----------

So, At first i got 40, while at second I got 50, taking i got 50.


Kindly, what do you think, where is my mistake? smile

#2 2016-09-05 20:06:12

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Integration as a series summaion

Hi;

Integration can be defined as:

b               b
∫ f(x).dx = ∑ f(x)
a               a

Who said integration and summation were the same thing? Sometimes, we can approximate one with the other but they are not the same thing.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#3 2016-09-05 20:13:19

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Integration as a series summaion


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#4 2016-09-05 20:27:57

peter010
Member
Registered: 2016-09-05
Posts: 22

Re: Integration as a series summaion

bobbym wrote:

Hi;

Integration can be defined as:

b               b
∫ f(x).dx = ∑ f(x)
a               a

Who said integration and summation were the same thing? Sometimes, we can approximate one with the other but they are not the same thing.

http://www.mathcentre.ac.uk/resources/Engineering%20maths%20first%20aid%20kit/latexsource%20and%20diagrams/8_12.pdf

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#5 2016-09-05 20:30:58

peter010
Member
Registered: 2016-09-05
Posts: 22

Re: Integration as a series summaion

thickhead wrote:

Thanks smile

So, what if I considered it like: 0 to 1 represents: 1, 1 to 2 represents: 2, 2 to 3 represents: 3, and so on, thus I can presume the following:

5
∫10 = (10*5) - (10*0) = 50
0

What do you think ? smile

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#6 2016-09-05 21:06:24

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Integration as a series summaion

As I said summation and integration can be used as approximations of each other. The discretizing trick will not always produce exact same answers.

For instance:


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#7 2016-09-05 21:16:36

peter010
Member
Registered: 2016-09-05
Posts: 22

Re: Integration as a series summaion

bobbym wrote:

As I said summation and integration can be used as approximations of each other. The discretizing trick will not always produce exact same answers.

For instance:

Thanks smile

I totally agree. Its here a special matter of integrating a constant, thus I presume exact answer could be achieved.

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#8 2016-09-05 22:54:09

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,273
Website

Re: Integration as a series summaion

Indeed, the integral is sometimes compared with a series to prove that it converges (or diverges). You can read about it here.

peter010 wrote:

Hello,

Integration can be defined as:

b               b
∫ f(x).dx = ∑ f(x)
a               a

peter010 wrote:

http://www.mathcentre.ac.uk/resources/Engineering%20maths%20first%20aid%20kit/latexsource%20and%20diagrams/8_12.pdf

That isn't the same as what you wrote -- you need the
(representing the width of the rectangles), otherwise the equality isn't true. Their definition of the definite integral was:

assuming the function is integrable on
to begin with. If you leave that out, then you are essentially approximating the definite integral of the function by rectangles of width 1, so in general there will be a discrepancy between the integral and the sum itself.

Last edited by zetafunc (2016-09-05 22:55:21)

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#9 2016-09-06 02:16:30

thickhead
Member
Registered: 2016-04-16
Posts: 1,086

Re: Integration as a series summaion


{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}

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#10 2016-09-06 02:37:21

zetafunc
Moderator
Registered: 2014-05-21
Posts: 2,273
Website

Re: Integration as a series summaion

He might do, yes, but the opening line of his post suggested that he was using that definition of an integral for a general function
.

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#11 2016-09-06 04:06:39

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Integration as a series summaion

He might do, yes, but the opening line of his post suggested that he was using that definition of an integral for a general function f.

That is correct and why I answered in the way I did.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#12 2016-09-06 06:10:07

peter010
Member
Registered: 2016-09-05
Posts: 22

Re: Integration as a series summaion

thickhead wrote:

I cant describe this case better than this smile smile

zetafunc wrote:

He might do, yes, but the opening line of his post suggested that he was using that definition of an integral for a general function
.

Thanks smile

bobbym wrote:

He might do, yes, but the opening line of his post suggested that he was using that definition of an integral for a general function f.

That is correct and why I answered in the way I did.

Thanks Indeed smile

Last edited by peter010 (2016-09-06 06:17:06)

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#13 2016-09-06 07:06:49

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Integration as a series summaion

Welcome to the forum.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#14 2016-09-06 09:15:52

peter010
Member
Registered: 2016-09-05
Posts: 22

Re: Integration as a series summaion

Here I put my code for calculating the integration by approximation method (the series),,,

//Integrating Functions
            double function;
            double width = .0001;
            double sum = 0, j = 1;
            int interval = 5;
            for (double i = width; j <= (interval / width); j++, i += width)
            {
                function = 10;
                sum += (width * function);
            }

            Console.WriteLine(sum);//50

            Console.WriteLine("\nDone!");


///

smile

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#15 2016-09-06 09:25:40

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Integration as a series summaion

Did you put a lot of very thin rectangles under the curve?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#16 2016-09-06 09:33:23

peter010
Member
Registered: 2016-09-05
Posts: 22

Re: Integration as a series summaion

bobbym wrote:

Did you put a lot of very thin rectangles under the curve?

This code is to meet any function precisely (not specifically the function of 10 ) with a rectangular width=0.0001 (it can be definitely adjusted by the user)

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#17 2016-09-06 09:44:26

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Integration as a series summaion

Hi;

Any function precisely? No amount of rectangles can do that unless it is an infinite number of them. There are even many more precise methods of numerical integration ( exactly what you are doing ) but each except for limited cases is just an approximation.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#18 2016-09-06 09:52:35

peter010
Member
Registered: 2016-09-05
Posts: 22

Re: Integration as a series summaion

bobbym wrote:

Hi;

Any function precisely? No amount of rectangles can do that unless it is an infinite number of them. There are even many more precise methods of numerical integration ( exactly what you are doing ) but each except for limited cases is just an approximation.

ok

i just wanted to share "my code" of series "approximation" method with ppl.

noth. more,,

Regards.

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#19 2016-09-06 09:55:36

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Integration as a series summaion

That is fine! You are making discoveries for yourself and thinking about how to get your computer to do them. The epitome of EM. I was just giving you some of what I do in that field. My favorites are numerical integration and curve fitting.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#20 2016-09-06 10:10:02

peter010
Member
Registered: 2016-09-05
Posts: 22

Re: Integration as a series summaion

bobbym wrote:

That is fine! You are making discoveries for yourself and thinking about how to get your computer to do them. The epitome of EM. I was just giving you some of what I do in that field. My favorites are numerical integration and curve fitting.

That is a lovely field.

And I appreciate and welcoming your advices, which is the reason why I come to this forum smile

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#21 2016-09-06 10:16:32

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Integration as a series summaion

Many integrals are what we call pathogenic ( diseased or sick ) in numerical work it means they do not respond well to numerical integration. One of these is

It is easy to do using calculus but not so easy using numerical methods. You might like to try it when you have some spare time.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

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#22 2016-09-06 10:36:54

peter010
Member
Registered: 2016-09-05
Posts: 22

Re: Integration as a series summaion

bobbym wrote:

Many integrals are what we call pathogenic ( diseased or sick ) in numerical work it means they do not respond well to numerical integration. One of these is

It is easy to do using calculus but not so easy using numerical methods. You might like to try it when you have some spare time.

hahaha I just did..

And I found yes it is diseased big_smile

Last edited by peter010 (2016-09-06 10:37:12)

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#23 2016-09-06 10:40:02

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Integration as a series summaion

Did you get anything close to 2 / 3 = .66666666...?


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

#24 2016-09-06 10:48:10

peter010
Member
Registered: 2016-09-05
Posts: 22

Re: Integration as a series summaion

bobbym wrote:

Did you get anything close to 2 / 3 = .66666666...?

= 0.266

but this still does not mean that i cannot customizing the solution/code to fit this kind of functions/

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#25 2016-09-06 10:55:18

bobbym
bumpkin
From: Bumpkinland
Registered: 2009-04-12
Posts: 109,606

Re: Integration as a series summaion

You can make particular formulas for any function but it appears there will always be some that will not like your formula.


In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

Offline

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