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".

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

height

- height equal to distance between the next points, the result becomes absolute. For now you didn't reach before and you use trapezes result will be approximate.

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:

- means two points (1=dx/dx, 1=dr/dr, 1=dl/dl) nearby and distance between them. It is one point increased by distance between them - an elementary piece:

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 (2013-01-08 14:11:57)

Give me the test.

bob bundy wrote:

Please choose the following

base radius of cone
height of cone
number of slices

Bob

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

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

.The circle area increased by distance to other circle not including other circle is

. 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 (2013-01-10 13:47:49)

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

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...

Then you could have something like

...

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)...

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...

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 (2013-01-12 12:08:41)

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

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