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**21122012****Member**- Registered: 2012-11-16
- Posts: 278

On this question:

"Why do you think this is ?"

you answered:

"The third row uses the trapezium rule to calculate (approximately) the area of each section".

**"The conditions imposed on functions, become a source of difficulties which will manage to be avoided only by means of new researches about the principles of integral calculus"**

** Thomas Ioannes Stiltes.** ...

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,913

I will try again.

Please choose the following

base radius of cone

height of cone

number of slices

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**21122012****Member**- Registered: 2012-11-16
- Posts: 278

"number of slices"

While you will have trapezes result will be approximate. As soon as you will take two areas of a circle

Bob

Try to translate by means of the dictionary start top from Russian. I here all accurately wrote all. I feel that my robot doesn't translate all sense of that that I want to inform you.

http://bolshoyforum.org/forum/index.php?topic=297286.0

I will try to explain a bean to you so. Present that on the plane there is circumference of any radius. ANY (! ! ! ) It is the line, it - not the area (! ! ! ) Now you to it add one more circumference in the same plane. Either it is more or has no value but such that between these two circumferences it was impossible to insert one more less.

It already area, instead of line. Elementary Square, the most smaller also is differential of the area of a circle. Any more line. But already area! It also is:

The sum of elementary segments of line (pieces) lying in one direction (not in parallel):

- is a line.

One point - One point it yet length not segment of line. Three points - not elementary. Integration is an absolute measure it doesn't depend on a unit of measure which the person can choose randomly.

You understand this?

P.S. I try to translate twice: from Russian into English then from English into Russian - sense is not adequate. I don't know what to do.

*Last edited by 21122012 (2013-01-07 15:11:57)*

**"The conditions imposed on functions, become a source of difficulties which will manage to be avoided only by means of new researches about the principles of integral calculus"**

** Thomas Ioannes Stiltes.** ...

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,913

<structural analysis> is true => <cone test is good>

So please do not give up on the test.

I will use upper and lower bounds.

sense is not adequate

I agree. Translation is one word at a time. Phrases are not translated properly. If a word has two meanings, translation may pick the wrong one.

Suggestion:

Give Russian and English. Stefy may help with meanings.

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**21122012****Member**- Registered: 2012-11-16
- Posts: 278

Give me the test.

**"The conditions imposed on functions, become a source of difficulties which will manage to be avoided only by means of new researches about the principles of integral calculus"**

** Thomas Ioannes Stiltes.** ...

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,913

Please choose the following

base radius of cone

height of cone

number of slices

Bob

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

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**21122012****Member**- Registered: 2012-11-16
- Posts: 278

bob bundy wrote:

Please choose the following

base radius of cone

height of cone

number of slicesBob

Didn't understand. You look post #28 the most top line.

** Thomas Ioannes Stiltes.** ...

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,913

I have made a new spreadsheet.

I have used R = 34

H = 17

number of slices = 17

Row 4 shows the r values from 0 to 34

Row 5 shows the h values from 0 to 17

Row 6 calculates the volume using

eg. The G column formula is =1/3*PI()*G4^2*G5

Row 7 calculates values of pi r^2. The formula for the G column is =PI()*G4^2

Row 9 calculates the area of a rectangle below the curve.

This is the yellow area shown in my second diagram. The formula in the G column is =G7*(H4-G4)

Row 10 sums these values. Formula in the G column is =SUM($B$9:G9)

Row 11 calculates the area of a rectangle above the curve.

This is the sum of the yellow and green areas in my diagram. The formula in the G column is =H7*(H4-G4)

Row 12 sums these values. Formula in the G column is =SUM($B$11:G11)

Row 10 gives the lower bound for the area under the curve. 44861

Row 12 gives the upper bound for the area under the curve. 53005

Therefore 44861 < area under curve < 53005

The correct volume of the cone is 20579

Conclusion: the following formula is NOT correct.

Bob

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**21122012****Member**- Registered: 2012-11-16
- Posts: 278

Bob!

You don't understand that such differential. It not SMALL INCREMENT. This ELEMENTARY INCREMENT! And you use the SMALL INCREMENT.

You didn't read post #28. Read, I there show a difference. But I now will repeat in relation to your example. You have the circle area. Not important what radius. Understand, radius length yet has no value! This area of a circle - the area, but not volume! This is element of Planimetrics, but not element of Solid Geometry. You understend me? This element of Planimetrics has its radius?

Now take the radius one point smaller or one point more and construct the circle area. Put from above on the first circle. Look at thickness sideways. It will be elementary volume of

. If you put just the same circle, the circle area increased by distance to other circle not including other circle is will be elementary volume of . Because Though eventually will reach only size Because

Now most important!

When you build cone volume with height equal to basis (!) radius that at distance from a beginning point (from top) at any distance on height with you will be placed the circle with the same radius (!) is a key to understanding of integration! In any point of height there will be element of Planimetrics in the form of a circle of the same radius. If height isn't equal to radius, and is its function, at each distance from top there will be a circle CORRESPONDING to value of function!

Therefore no trapezes will exist! Trapezes are delusions (errors) of Calculus.

They have no place to undertake. They won't appear anywhere because there will be no two distances from top on which there will be identical circles!

There will be only elementary truncated cones.

In what difference between

and

Which in the first case creates volume a cone, and in the second case creates cylinder volume???!!!

Everything is concluded in a difference of these two expressions:

prompts that at distance from top circle will settle down. prompts that at any distance from top IDENTICAL circles: will settle down.You understand?

P.S.

In everything the limit which uses Calculus is guilty.

Because this limit it is possible to give only presentation of VALUE of the DERIVATIVE but not the most derivative.

In Structural Analysis the absolute limit on accuracy for receiving derivative function instead of its value is used:

*Last edited by 21122012 (2013-01-09 14:47:49)*

** Thomas Ioannes Stiltes.** ...

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,913

I do not understand why you use calculus at all in your structural analysis.

You claim it has errors and then use it anyway.

I have read post 28.

Integral calculus is built upon small increments. If you won't accept that then I suggest you remove all references to calculus from your theory and present it properly.

I do not understand the term elementary increment. I think your translation robot has failed to give you the correct English.

I have already tried to explain to you why

fails to give you the correct result. Of course you get the volume of a cylinder if you treat pi r squared as a constant. Most mathematicians know you have to change the 'r' term to a function of 'h' before you integrate.

I see no point in continuing until you clear up what you are trying to do in post 22.

Now I look at post 22 again I see that your integration for a cylinder is wrong too.

Please use the upper bound / lower bound method to justify these or change them.

Bob

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**21122012****Member**- Registered: 2012-11-16
- Posts: 278

I translate from English. I can't understand that you want. What mine have to be actions?

** Thomas Ioannes Stiltes.** ...

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,913

hi 21122012

Here you say

How can this be from o to r ? The variable is h.

Note:

And you say

Note:

This is my last word on integration.

Bob

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,919

bob bundy wrote:

That is not correct. You cannot have h both in the integration limits and as the variable which you are integrating with respect to...

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,544

Hi anonimnystefy;

That is not correct. You cannot have h both in the integration limits and as the variable which you are integrating with respect to...

Of course you can.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,919

Then you could have something like

...Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,544

Hi;

Take these over to alpha.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,919

Alpha understand it differently then they should be understood. It looks at the two k's (and x's) as different, while they are in fact the same k (x)...

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,544

No, they are not the same, in the integral you do not substitute for the x in the dx. The index of summation obviously is the same.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,919

Hi bobbym

All instances of a variable name in a single expression represent a single variable, so the three x's and k's are the same...

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 94,544

One thing at a time.

I do not see any reason why this

is not allowed.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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**21122012****Member**- Registered: 2012-11-16
- Posts: 278

bob bundy wrote:

hi 21122012

Here you say

How can this be from o to r ? The variable is h.

Hi bob!

WOW!!!!!!!!

Look this:

It is a special case of a general view:

bob bundy wrote:

Note:

It is a special case. You choose only one option from all possible options of height. Such, when height равнв to basis radius. I didn't think that it is difficult for understanding.

bob bundy wrote:

And you say

I couldn't say such nonsense. It not cone volume, because it cylinder volume.

bob bundy wrote:

Note:

This is my last word on integration.

Bob

You don't understand difference of a variable from value of a variable which is constants.

If you took two independent

variables, then made their dependent

Then took values of these dependent variables

these values can't become independent variables

It is absurdity!

Function of a type:

in a geometrical form of a mnterpretation where - x radius, y - height can be constructed only in the form of cylinder volume.

The volume of cone can be constructed only if

Excuse don't take offense at me but with such representation of integration as at you it is impossible to accuse me of mistakes. It is frivolous. If you saw it as someone somewhere goes figures in chess you will speak to the grand master that you will win against him in advance. Once again I am sorry but you told the first that I am mistaken. I am not mistaken. I can teach as to do correctly those who is mistaken.

*Last edited by 21122012 (2013-01-11 13:08:41)*

** Thomas Ioannes Stiltes.** ...

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**21122012****Member**- Registered: 2012-11-16
- Posts: 278

anonimnystefy wrote:

bob bundy wrote:That is not correct. You cannot have h both in the integration limits and as the variable which you are integrating with respect to...

This isn't an error. It is an unnecessary duplicator of a variable of a mntegrirovaniye. In the main theorem of calculation it is a case when the variable doesn't lie in an interval and is the interval end.

*Last edited by 21122012 (2013-01-11 17:56:29)*

** Thomas Ioannes Stiltes.** ...

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,913

hi 21122012,

**A demonstration that 2 variables may be independently chosen and yet, still be related.**

You wrote:

You don't understand difference of a variable from value of a variable which is constants.

If you took two independent variables, then made their dependent

Then took values of these dependent variables these values can't become independent variables

There are many formulas with three variables which will allow you to choose two variables 'independently'. **The three are still related.**

example 1

You may choose D = 24 Km, and S = 4 Km/hr. This was a free and independent choice for D and S. But they are still related by the formula.

example 2

is the equation for a circle, centred on (0,0)

You may choose any values for x and y because any point will lie on some circle.

But x and y are related.

example 3

Once again you may choose u and t independently but the formula still holds.

Why are you so reluctant to accept that, for a cone, there is a formula connecting r and h?

If r is fixed, then it is not a cone is it?

r has to vary as h varies to make a cone.

Bob

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**21122012****Member**- Registered: 2012-11-16
- Posts: 278

Hy bob

You forgot about what we speak. More true you forgot that moment because of which our dialogue began. Therefore you displaced sense in other party. I will remind you and you will see that you lost the conversation reason.

"....I will show one real mistake. But usually after such my subjects in Russia deleted at once. I will try here. We look the link:

http://en.wikipedia.org/wiki/Partial_derivative

We see a formula of a full derivative of volume of a cone on height:

We integrate this derivative and we receive... cylinder volume:

...."You remembered why we started talking about a cone and the cylinder? Therefore when we tell about volumes of these two geometrical figures that I always I speak about them and I remember the reason for which we speak about it. And you tell everything that doesn't treat at all a subject of our dispute. You didn't prove to me that a formula

it is cone volume. And still didn't give integral for calculation of volume of the cylinder which would differ from this formula THOUGH SOMETHING!

*Last edited by 21122012 (2013-01-12 08:10:17)*

** Thomas Ioannes Stiltes.** ...

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**bob bundy****Moderator**- Registered: 2010-06-20
- Posts: 6,913

You didn't prove to me that a formula

it is cone volume.

Yes I did. But it is so clear in my head I will do it again.

Obviously h = r = 0 gives C = 0

So

The reason you get the formula for a cylinder is because you insist on saying pi r^2 is constant.

Of course it isn't for a cone, but it is for a cylinder.

Note: This work has nothing to do with partial derivatives. As r and h are related it is possible to avoid any parrtial derivatives at all simply be using the relationship between r and h.

If you will not accept this, then we might as well cease communicating.

Bob

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