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## #1 2005-11-07 23:15:59

szk_kei
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### Integral relationship

Relationship between the integral of the function and
the integral of its inverse function
INTGR(y)dx + INTGR(x)dy = xy + C

http://www.freewebs.com/keiichi_suzuki/

## #2 2005-11-08 07:23:47

MathsIsFun

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### Re: Integral relationship

BTW, you can use the "∫" symbol (I have those symbols just under the forum title - just drag you mouse across one, copy then paste, and you can get: ∫y dx + ∫x dy)

So ... why does ∫y dx + ∫x dy = xy + C ??

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

## #3 2005-11-08 07:57:58

mathsyperson
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### Re: Integral relationship

I tried to look at szk_kei's proof, but when I went to his site and clicked the link to it I just got an acrobat reader thing with 4 blank pages. Probably my computer doesn't support something that it needs to. I tried it out though, and it seems to work.

y = x

∫ydx = 0.5x² + c
∫xdy = 0.5y² + c

∫ydx + ∫xdy = 0.5x² + 0.5y² + c

x = y ∴ x² = y² ∴ 0.5x² + 0.5y² + c = x² + c
x = y ∴ x² + c = xy + c

Last edited by mathsyperson (2005-11-08 07:58:09)

Why did the vector cross the road?
It wanted to be normal.

## #4 2005-11-08 17:24:23

MathsIsFun

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### Re: Integral relationship

Graphically ∫ydx is the area "below" the curve to the x axis, while ∫xdy is the area to the "left" of the curve to the y-axis, so together they form a square (well they do if the limits are 0 to some value).

So this probably applies to y = x², too, and many more functions ...

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

## #5 2005-11-08 21:33:14

szk_kei
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### Re: Integral relationship

y=f(x):differentiable

(xy)'=y+x(dy/dx)
xy+C=∫ydx + ∫x(dy/dx)dx
=∫y dx + ∫x dy

If you want to sea its detail, please visit my site http://www.freewebs.com/keiichi_suzuki/
->[The Study (Mathematics created by myself)]

## #6 2005-11-09 05:22:48

kylekatarn
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### Re: Integral relationship

You can find those relations in all calculus books.

## #7 2005-11-09 23:57:06

szk_kei
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### Re: Integral relationship

When I show an expression without proof, they say always "It can never hold".
When I show an expression with proof, someone says "It is only an application of integration by parts",
someone says "It is simple and beautiful".

## #8 2005-11-10 05:07:18

mathsyperson
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### Re: Integral relationship

#### MathsIsFun wrote:

Graphically ∫ydx is the area "below" the curve to the x axis, while ∫xdy is the area to the "left" of the curve to the y-axis, so together they form a square (well they do if the limits are 0 to some value).

That's a very nice way of putting it without getting involved in lots of heavy maths. It actually forms a rectangle though.

Last edited by mathsyperson (2005-11-10 05:07:26)

Why did the vector cross the road?
It wanted to be normal.

## #9 2005-11-10 23:24:01

szk_kei
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### Re: Integral relationship

Your definition is monotone increasing function passing through the origin and  differentiable.
My definition is only differentiable.

## #10 2005-11-11 07:22:11

MathsIsFun

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### Re: Integral relationship

The y=x function was only an example, I think

We could try another example: y=x²

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

## #11 2005-11-11 08:16:13

mathsyperson
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### Re: Integral relationship

Yes, it was. I'll do the y = x², because it gives me a chance to practice the new code things.

y = x²
∫ydx = x³/3 + c

x=√y
∫xdy = 2/3√(y³) + d

x³/3 + c + 2/3√(y³) + d = x³/3 + 2/3x³ + c + d = x³ + c + d

x³ = x*x² = xy + c + d

c and d are both arbitrary constants, so they can be combined into an arbitrary constant C.

∫xdy + ∫ydx = xy + C. It still works!

Last edited by mathsyperson (2005-12-03 03:49:16)

Why did the vector cross the road?
It wanted to be normal.

## #12 2005-11-14 22:42:26

szk_kei
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### Re: Integral relationship

My name is Keiichi Suzuki.
I'm a 48-year-old male.
I'm a Japanese.

I publish online mathematics created by myself,
including arithmetic (mathematics) for children.
http://www.freewebs.com/keiichi_suzuki/

My site is listed in
http://dir.yahoo.com/Science/Mathematics/Calculus/

## #13 2005-11-14 23:06:35

ganesh
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### Re: Integral relationship

Hi Mr.Keiichi Suzuki,
Welcome to the forum.

Character is who you are when no one is looking.

## #14 2005-11-15 00:25:54

szk_kei
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### Re: Integral relationship

I'm not mathematician.
I'm working for a manufacturer.

## #15 2005-11-15 17:20:48

ganesh
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### Re: Integral relationship

I'm not a mathematician either.
I work taxing manufacturers (collecting taxes).

Character is who you are when no one is looking.

## #16 2005-11-15 19:07:54

MathsIsFun

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### Re: Integral relationship

LOL! Thank goodness you guys live in different countries!!

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

## #17 2005-11-16 04:07:17

kylekatarn
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### Re: Integral relationship

fortunately for me, I'm still a student. : )

## #18 2005-11-18 01:46:11

szk_kei
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### Re: Integral relationship

There are some topics in my site.

Taylor Series.
What is the necessary and sufficient condition for the intervals of n of  the degree as expressed below ?
f(x)=A0 + An(x-a)^n + A2n(x-a)^2n + A3n(x-a)^3n + A4n(x-a)^4n + +

http://www.freewebs.com/keiichi_suzuki/

## #19 2005-12-02 11:54:03

szk_kei
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### Re: Integral relationship

Can you see the full explanation of integral relation and property of laurent expansion with an acrobat reader.

http://www.freewebs.com/keiichi_suzuki/

## #20 2005-12-28 11:50:25

szk_kei
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### Re: Integral relationship

Can you see the full explanation of integral relation and property of laurent expansion with an acrobat reader.

## #21 2005-12-28 17:11:07

MathsIsFun

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### Re: Integral relationship

Looks good, szk_kei, though I haven't checked if it is right or not.

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman

## #22 2005-12-29 02:16:27

krassi_holmz
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### Re: Integral relationship

Very good site, Keiichi.

IPBLE:  Increasing Performance By Lowering Expectations.

## #23 2005-12-29 02:21:47

krassi_holmz
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### Re: Integral relationship

And we can form something like that for definite integrals:

Last edited by krassi_holmz (2005-12-29 02:25:27)

IPBLE:  Increasing Performance By Lowering Expectations.

## #24 2005-12-29 02:36:18

krassi_holmz
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### Re: Integral relationship

Geometric proof

IPBLE:  Increasing Performance By Lowering Expectations.

## #25 2005-12-29 19:46:35

szk_kei
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### Re: Integral relationship

>Looks good, szk_kei, though I haven't checked if it is right or not.

Thank you