Math Is Fun Forum

Hi bob!

Here initial postulate of STRUCTURAL ANALYSIS:

I don't decide to translate it into English not to lose meaning.

Hi to All
The Fundamental Theorem of Calculus proves that:
1. That differentiation and antidifferentiation are essentially inverse processes:

2. How to evaluate a definite integral using the antiderivative of the integrand:

where F is an antiderivative of f (i.e., F' = f).
I think that on it the correct theory comes to an end.

Nobody proved it and it doesn't make sense.

Because:

You didn't explain to me as it is possible using to receive ONE formula TWO various answers one of which is 3 times more than another.

This is THREE volumes !!!

So in Structural Analysis.

Write as it looks at you:

Yes but it not I am the robot so translates.
Here so it translated your phrase:

On English: "I think I finally understand what he means by schedule- grap"

->

On Russian: "Я думаю, что наконец понимаю то, что он подразумевает графиком - граф!"

Identical type of expressions - decide differently. On the ode of a variable the derivative undertakes, other letter registers.

Two functions, each function of two variables, in one option dependent are given, in the other - of the independent - is unclear on what function to consider a total derivative...

This is cardsharpering instead of exact science.

Hi Bob!

You recognize an error of calculus which calls cylinder volume as volume a cone or give me a formula of uncertain integral for cylinder volume. But there can be at you such science which can't give a formula of volume of the cylinder? Then your science is necessary to nobody!

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!

Hi bob!

WOW!!!!!!!!

Look this:

It is a special case of a general view:

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.

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

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.

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:

Give me the test.

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".

Hi Bob

Sense of  written in post#8 - is represented on the chart in post #14 !!!

In rectangular system of coordinates the segment of line:

image as a function increment:

in the form of the sum:

of identical points is used:

The distance between the next points will be equal:

In the Cartesian system of coordinates the distance between the next points equal is used

then points from which the segment of line consists are:

Points of a piece are inadequate and matter equal to value of derivatives at argument change.

Bob, look this animation. When on the right the green segment of line

increases in a size at one a point up at the same time on the right its equivalent the area of a circle increases by one circumference and so up to the end: more and more points in a segment of line - more and more circumferences in a circle. Circumferences of the various size - a points of various value. In the beginning the circumference is equal to zero - the point is equal to zero. At the end the circle is equal to a maximum - a point same.

Part second. Geometrical and graphic interpretation of concept a derivative on the example of function

If it is badly visible it is possible to look here.

What exactly do you want to learn about this integral? It is a half of the area of a square.

The derivative of this area will be the party of this square.