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Topic: integral and integration #6 Textbook 6th ed.: TRUE CALCULUS; without
the phony limit concept

Replies: 0

 plutonium.archimedes@gmail.com Posts: 18,572 Registered: 3/31/08
integral and integration #6 Textbook 6th ed.: TRUE CALCULUS; without
the phony limit concept

Posted: Jun 17, 2013 5:13 AM

First let me correct an error in the previous post #5 where I said
this:

In True Calculus, we have no
"pathological functions" such as y = sin(1/x) or the Weierstrass
function. In True Calculus those would be continuous, differentiable
and integrable everywhere.

That is a error on my part for it does not take into account division
by 0 is a point of discontinuity. So for y = sin(1/x) or the function
y= 1/x there is no derivative and no integral in the cell of 0 to 0.1
in 10-Grid because there is no y value when x=0. Now I need to define
"cell" in Calculus.

Alright, this is going to be the easiest of Calculus to teach, the
integral and integration.

I say that because it is simply the summation of picketfences over an
interval of the x-axis, or I should say the summation of areas in each
cell of the function graph. Let me include a term, new term in
Calculus-- "the cell", for the width of the unit picketfence which in
10-Grid is 0.1 width.

.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
0  .1   .2  .3  .4  .5   .6  .7  .8  .9  1.0  -->

Then there are 10 cells there with the first cell being 0 to .1 wide
and then going up along the y axis. The second cell would be from 0.1
to 0.2 and then up along the y axis.

There is one key idea in integrals and integration that involves set
theory. All intervals in Calculus are closed intervals. For example in
the 10 Grid system we cannot have open intervals of say (0, 0.2)
because we have those gaps of empty space between successive number-
points. So what does (0,0.2) mean in set theory of an open interval?
It would mean the set {0.1} containing a single member of 0.1. The
closed interval [0, 0.2] has three members of the set {0, 0.1, 0.2}.
So in True Calculus, set theory has no open intervals, no full open
intervals and no half open intervals because we cannot put a gap of
empty space into set theory. Some of the gaps of empty space are more
than 10 metric distance long, for example the derivative of the
sawtooth function is the hypotenuse of the right triangle:

F(x) = 0 when x is even number and F(x) = 10 when x is odd number. So
the graph of this Sawtooth function in 10-Grid, 1st quadrant only,
looks like this:

.    x    .   x    .   x    .   x    .    x   . 10.0
.    .    .    .    .    .    .    .    .    .    .  9.9
.    .    .    .    .    .    .    .    .    .    .  9.8
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
out of scale for there should be 10 blocks of 10
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
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.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
.    .    .    .    .    .    .    .    .    .    .
x    .   x    .    x   .    x    .   x    .   x
0  .1   .2  .3  .4  .5   .6  .7  .8  .9  1.0  -->

So in True Calculus, all the intervals are closed intervals.

And for the sawtooth function above, the integral is the summation of
those steep triangles in every one of those successive number points
of width 0.1 metric, for every cell has one steep triangle. So the
integral is the area of the triangle in the 0 to 0.1 interval added
with the triangle in 0.1 to 0.2 interval, and so on. In fact, the
integral of this sawtooth function is 1/2 of (10 by 10) or 50 square
units. For it is apparent that if we stack the triangles of one cell
with its successive cell we have a rectangle that is half the area of
10x10.

Many integrals will be the summation of triangles only, while many
others will be the summation of picketfences such as the y = x
function. Some functions will be the summation of pure rectangles only
for the integral, such as the function y= 3.

The functions, y= 1/x or y = x^2 will be picketfence integrals, since
each cell of 0.1 width will be partly a rectangle with a triangle
atop.

Now notice something strange about the function y= 1/x of its
integration, in that it has no y value when x=0 because division by
zero is undefined. So in that cell of 0 to 0.1 we have no area to add
and we have no derivative to draw either. So with the function y= 1/x
we say it is discontinuous function at x=0, and start the integral and
integration at 0.1 and pretend as though that is the start of the
function.

--

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Archimedes Plutonium
http://www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies