Rensselaer,Dr.Donald Schwendeman,Dr.Jeffrey Banks,Dr.Kristin Bennett, do you let Kibo flood sci.math with spam and flood sci.physics with these pretend Russian-USA discourse???
Kibo on book and propaganda spam flooding sci.math,sci.physics
Rensselaer Polytechnic Institute Physics dept Dr.Martin Schmidt (ee), Dr.Ivar Giaever
Dr.Vincent Meunier, Dr.Ethan Brown,Dr.Glenn Ciolek is Kibo Parry (Volney) running this spam machine out of Rensselaer, flooding sci.math and sci.physics with Indonesia slot spam and pretend Russian propaganda. Given that rpi dot edu address of Kibo.
Large Primes
by
James 'Kibo' Parry
May 26, 1989, 10:35:51 AM
What's the largest prime currenlty known? (All the information
I could dig up here was either fairly old or contradictory...)
james "kibo" parry | Some days you just can't get rid of a bomb. kibo%
pawl.r...@itsgw.rpi.edu (internet)
userfe0n@rpitsmts (bitnet) | Anything I say represents the opinion of kibo%
mts.r...@itsgw.rpi.edu | myself and not this computer.
Dr.Bulent Yener,Dr.Donald Drew,Dr.William Siegmann, Rensselaer Polytech is this your spam??
Kibo is this your and Rensselaer spam-- flood of slot machine and other
Kibo Parry Moroney Volney wrote:
On Wednesday, December 6, 2017 at 12:30:22 AM UTC-6, Michael Moroney wrote:
Silly boy, that's off by more than 12.6 MeV, or 12% of the mass of a muon. Hardly "exactly" 9 muons.
Wednesday, December 6, 2017 at 9:52:21 AM UTC-6, Michael Moroney wrote: Or, 938.2720813/105.6583745 = 8.88024338572. A proton is about the mass
of 8.88 muons, not 9. About 12% short.
Why Volney?? Because they stop short of completing the Water Electrolysis Experiment by only looking at volume, when they are meant to weigh the mass of hydrogen versus oxygen?? Such shoddy minds in experimental physics and chemistry.
Rensselaer Polytechnic Institute Physics dept Dr.Martin Schmidt (ee), Dr.Ivar Giaever
Vincent Meunier, Ethan Brown, Glenn Ciolek, Julian S. Georg, Joel T. Giedt, Yong Sung Kim, Gyorgy Korniss, Toh-Ming Lu, Charles Martin, Joseph Darryl Michael, Heidi Jo Newberg, Moussa N'Gom, Peter Persans, John Schroeder, Michael Shur, Shawn-Yu Lin,
Humberto Terrones, Gwo Ching Wang, Morris A Washington, Esther A. Wertz, Christian M. Wetzel, Ingrid Wilke, Shengbai Zhang
Rensselaer math department
Donald Schwendeman, Jeffrey Banks, Kristin Bennett, Mohamed Boudjelkha, Joseph Ecker, William Henshaw, Isom Herron, Mark H Holmes, David Isaacson, Elizabeth Kam, Ashwani Kapila, Maya Kiehl, Gregor Kovacic, Peter Kramer, Gina Kucinski, Rongjie Lai,
Fengyan Li, Chjan Lim, Yuri V Lvov, Harry McLaughlin, John E. Mitchell, Bruce Piper, David A Schmidt, Daniel Stevenson, Yangyang Xu, Bulent Yener, Donald Drew, William Siegmann
Kibo loves TEACHING TRUE MATHEMATICS for Xmas stocking stuffers.
Why Dr.Wiles, Wolfgang Mueckenheim fail math-- conic sections, calculus. Are they playing slots and not enough math study??
Kibo Parry Volney, if Dr. Tao had studied under TEACHING TRUE MATHEMATICS, would he have had more commonsense to know slant cut of cylinder is ellipse, but not cone for its asymmetry makes the slant cut a Oval, not ellipse.
On Tuesday, September 26, 2023 at 4:21:58 PM UTC-5, Volney wrote:
The punishment will continue until morale improves.
TEACHING TRUE MATHEMATICS, AP seeks the super easiest calculus possible on Earth-- polynomials as the only valid functions-- thus, and therefore, making derivative and integral as easy as Power Rule- 14 year olds master calculus. Because the Power Rule
is merely add or subtract 1 from exponent so we can teach calculus in High School.
Only Math textbooks with the true numbers of mathematics-- Decimal Grid Numbers, not the insane silly Reals & Complex with their crank crackpot imaginary b.s.
I doubt the two math cranks Andrew Wiles and Terence Tao will ever understand mathematics for they continue to refuse to admit to even the most simple truths of mathematics-- slant cut of cone is Oval, not ellipse. A cylinder slant cut is ellipse, never
cone.
Kibo Parry Volney, if you had studied under TEACHING TRUE MATHEMATICS, probably today would understand what a correct percentage was instead of your failureship. And likely Dr. Wiles if not blind in his eyes had studied under TEACHING TRUE MATHEMATICS,
would know slant cut of cylinder is truly a ellipse but not of cone for that slant cut is a oval.
Old Math is in a world of hurt for it does not even have the correct numbers of mathematics. Old Math was arrogant and ignorant starting year 1900 when Quantum Mechanics in physics took off and it means the world is discrete and not continuous. Yet the
foolish bozos of Old Math stuck with their continuous Reals and even had the idiotic notion of going further out on the limb of madness with Cohen's continuum hypothesis, while Quantum Mechanics gave us a new age in physics with their discrete world. One
would think the idiots of Old Math would finally look at physics and pay attention and learn something. No. They never did. And so today in October of 2023 we still have idiots of math teaching calculus with never a valid proof of Fundamental Theorem of
Calculus, because Reals are not the true numbers of mathematics, the Decimal Grid Number System is the true numbers of math for they are discrete, and they make calculus, a billion, perhaps a trillion times easier to study , to learn to understand. In
fact, we TEACHING TRUE MATHEMATICS, teaches calculus to 13 and 14 year olds. Because calculus is as easy as add or subtract 1 from the exponent.
TEACHING TRUE MATHEMATICS the fake calculus of Thomas Hales, Andrew Wiles, Ken Ribet, Ruth Charney, Terence Tao, John Stillwell with their fake "limit analysis" for a true proof of Fundamental Theorem of Calculus (FTC) has to be a geometry proof for the
integral is area under a graphed function. This is why only a polynomial can be a valid function of math, for the polynomial is a function of the straightline Y --> mx + b. All the other so called functions have no straightline-- they are curves of
continuum and cannot give a proof of the fundamental theorem of calculus.
The proof of FTC needs a empty space Discrete Geometry from one point to the next point so as to allow for the construction of a midpoint between point A to point B and thus to hinge up from A at the midpoint and to determine the next point B in the
derivative. This is why Calculus is so enormously a tool for physics, as point A predicts point B.
Discrete Geometry is required for the proof of FTC and that requires the true numbers of mathematics be Decimal Grid Numbers, for they cannot be the continuum idiocy of Reals and Complex.
To make a half circle function in True Math, we have to go out to something like 10^6 Grid to make the points close enough together for the function visual to start looking like a half circle. But still there are holes in between one point and the next
point to allow the existence of calculus.
On a downward slope function, we have a different graphics than the usual upward slope function. For the upward slope requires the midpoint in the empty space to predict the next point of the thin rectangle that occupies that empty space (see the
graphics below and in my books TEACHING TRUE MATHEMATICS). In a downward slope function graph we still have those thin rectangles occupy the empty space for integral but we do not need to construct the midpoint, we simply shave away a right triangle that
reveals-- predicts point B starting from point A on the other side.
TEACHING TRUE MATHEMATICS, AP seeks the super easiest calculus possible on Earth-- polynomials as the only valid functions-- thus, and therefore, making derivative and integral as easy as Power Rule- 14 year olds master calculus. Because the Power Rule
is merely add or subtract 1 from exponent so we can teach calculus in High School.
Old Math makes and keeps Calculus as classroom torture chambers with their 1,000s of different functions yet the polynomial is the only valid function of math, and makes it super super easy to learn calculus
TEACHING TRUE MATHEMATICS, AP seeks the super easiest calculus possible on Earth-- polynomials as the only valid functions-- thus, and therefore, making derivative and integral as easy as Power Rule- 14 year olds master calculus.
If you come to me with a pathetic non polynomial especially that ugly trig functions, I have you go home and convert your nonsense to a polynomial. The Lagrange interpolation converts stupid nonfunctions like trig, into valid functions of polynomials.
TEACHING TRUE MATHEMATICS textbooks, makes calculus as easy as adding or subtracting 1 from exponent--only valid functions are polynomials contrast with mainstream--vomiting during exams, torture chamber and nervous breakdown by sado-masochist teachers.
Old Math is thousands of different kook functions with thousands of different rules. AP Calculus is one function-- the polynomial for we care about truth in math, not on whether kooks of math become rich and famous off the suffering-backs of students put
through a torture chamber that is present day calculus. If you come to math with a function that is not a polynomial, you have to convert it to a polynomial. Once converted, calculus is super super easy. But math professors seem to enjoy torturing
students, not teaching them. Psychology teaches us that when a kook goes through a torture chamber and comes out of it as a math professor-- they want to be vindictive and sado masochists and love to torture others and put them through the same torture
chamber that they went through. AP says-- stop this cycle of torture and teach TRUE CORRECT MATH.
TEACHING TRUE MATHEMATICS textbooks, makes calculus as easy as adding or subtracting 1 from exponent--only valid functions are polynomials contrast with mainstream--vomiting during exams, torture chamber and nervous breakdown by sado-masochist teachers.
Old Math is thousands of different kook functions with thousands of different rules. AP Calculus is one function-- the polynomial for we care about truth in math, not on whether kooks of math become rich and famous off the suffering of students put
through a torture chamber that is present day calculus. If you come to math with a function that is not a polynomial, you have to convert it to a polynomial. Once converted, calculus is super super easy. But math professors seem to enjoy torturing
students, not teaching them.
Old Math calculus textbooks like Stewart are more than 1,000 pages long and they need that because they have a mindless thousand different functions and no valid proof of Fundamental Theorem of Calculus. AP's calculus is less than 300 pages, because we
have a valid geometry proof of Fundamental Theorem of Calculus which demands the only valid function of math be a polynomial function. We can teach calculus in Junior High School for the calculus is reduced to adding or subtracting 1 from the exponent.
The only hard part of calculus in New Math is to convert the boneheaded function into a polynomial that was brought to the table by the boneheaded math professor who thinks that a function does not need to be a polynomial.
AP calculus transforms the calculus classroom. It is no longer vomiting during exams. No longer a torture chamber for our students of youth, and no longer a nightmare nor nervous breakdown for our youthful students, who, all they ever wanted was the
truth of mathematics.
Teaches that derivative predicts next point of function graph--silly Old Math has derivative as tangent to function graph unable to predict. The great power of Calculus is integral is area under function graph thus physics energy, and its prediction
power of the derivative to predict the next future point of function graph thus making the derivative a "law of physics as predictor". Stupid Old Math makes the derivative a tangent line, while New Math makes the derivative the predictor of next point of
function graph. No wonder no-one in Old Math could do a geometry, let alone a valid proof of Fundamental Theorem of Calculus, for no-one in Old Math even had the mind to realize Calculus predicts the future point in the derivative.
TEACHING TRUE MATHEMATICS-- only math textbooks with a valid proof of Fundamental Theorem of Calculus--teaches that derivative predicts next point of function graph--silly Old Math has derivative as tangent to function graph unable to predict. This is
why calculus is so important for physics, like a law of physics-- predicts the future given nearby point, predicts the next point. And of course the integral tells us the energy. Silly stupid Old Math understood the integral as area under the function
graph curve, but were stupid silly as to the understanding of derivative-- predict the next point as seen in this illustration:
From this rectangle of the integral with points A, midpoint then B
______
| |
| |
| |
---------
To this trapezoid with points A, m, B
B
/|
/ |
m /----|
/ |
| |
|____|
The trapezoid roof has to be a straight-line segment (the derivative)
so that it can be hinged at m, and swiveled down to form rectangle for integral.
Or going in reverse. From rectangle, the right triangle predicts the next successor point of function graph curve of B, from that of midpoint m and initial point of function graph A.
My 134th published book
Introduction to TEACHING TRUE MATHEMATICS: Volume 1 for ages 5 through 26, math textbook series, book 1 Kindle Edition
by Archimedes Plutonium (Author)
The 134th book of AP, and belatedly late, for I had already written the series of TEACHING TRUE MATHEMATICS in a 7 volume, 8 book set. This would be the first book in that 8 book set (one of the books is a companion book to 1st year college). But I
suppose that I needed to write the full series before I could write the Introduction and know what I had to talk about and talk about in a logical progression order. Sounds paradoxical in a sense, that I needed to write the full series first and then go
back and write the Introduction. But in another sense, hard to write an introduction on something you have not really fully done and completed. For example to know what is error filled Old Math and to list those errors in a logical order requires me to
write the full 7 volumes in order to list in order the mistakes.
Cover Picture: Mathematics begins with counting, with numbers, with quantity. But counting numbers needs geometry for something to count in the first place. So here in this picture of the generalized Hydrogen atom of chemistry and physics is a torus
geometry of 8 rings of a proton torus and one ring where my fingers are, is a equator ring that is the muon and thrusting through the proton torus at the equator of the torus. So we count 9 rings in all. So math is created by atoms and math numbers exist
because atoms have many geometry figures to count. And geometry exists because atoms have shapes and different figures.
Product details
• ASIN : B08K2XQB4M
• Publication date : September 24, 2020
• Language : English
• File size : 576 KB
• Text-to-Speech : Enabled
• Screen Reader : Supported
• Enhanced typesetting : Enabled
• X-Ray : Not Enabled
• Word Wise : Not Enabled
• Print length : 23 pages
• Lending : Enabled
• Best Sellers Rank: #224,974 in Kindle Store (See Top 100 in Kindle Store) ◦ #3 in 45-Minute Science & Math Short Reads
◦ #23 in Calculus (Kindle Store)
◦ #182 in Calculus (Books)
#5-2, My 45th published book.
TEACHING TRUE MATHEMATICS: Volume 2 for ages 5 to 18, math textbook series, book 2
by Archimedes Plutonium (Author) (Amazon Kindle edition)
Last revision was 2NOV2020. And this is AP's 45th published book of science.
Preface: Volume 2 takes the 5 year old student through to senior in High School for their math education.
This is a textbook series in several volumes that carries every person through all his/her math education starting age 5 up to age 26. Volume 2 is for age 5 year old to that of senior in High School, that is needed to do both science and math. Every
other math book is incidental to this series of Teaching True Mathematics.
It is a journal-textbook because Amazon's Kindle offers me the ability to edit overnight, and to change the text, almost on a daily basis. A unique first in education textbooks-- almost a continual overnight editing. Adding new text, correcting text.
Volume 2 takes the 5 year old student through to senior in High School for their math education. Volume 3 carries the Freshperson in College for their math calculus education.
Cover Picture: The Numbers as Integers from 0 to 100, and 10 Grid when dividing by 10, and part of the 100 Grid when dividing by 100. Decimal Grid Numbers are the true numbers of mathematics. The Reals, the rationals & irrationals, the algebraic &
transcendentals, the imaginary & Complex, and the negative-numbers are all fake numbers. For, to be a true number, you have to "be counted" by mathematical induction. The smallest Grid system is the Decimal 10 Grid.
Product details
ASIN : B07RG7BVZW
Publication date : May 2, 2019
Language : English
File size : 2024 KB
Text-to-Speech : Enabled
Screen Reader : Supported
Enhanced typesetting : Enabled
X-Ray : Not Enabled
Word Wise : Not Enabled
Print length : 423 pages
Lending : Enabled
Best Sellers Rank: #235,426 in Kindle Store (See Top 100 in Kindle Store)
#15 in General Geometry
#223 in Geometry & Topology (Books)
Alright I come to realize I have no graphic explanation for the proof of the Fundamental Theorem of Calculus for a downward slope function graph. I gave a proof for the upward slope function.
We start with the integral rectangle in the Cell, a specific cell of the function graph. In 10 Decimal Grid there are exactly 100 cells for each number interval, say from 0 to 0.1, then the next cell is 0.1 to 0.2. The midpoint in each cell belongs to a
number in the next higher Grid System, the 100 Grid. So the midpoint of cell 1.1 to 1.2 is 1.15 as midpoint.
Now the integral in that cell of 1.1 to 1.2 is a rectangle and say our function is x^2 --> Y. So the function graph is (1.1, 1.21) and (1.2, 1.44). Now we are strictly in 10 Grid borrowing from 100 Grid.
So say this is our Integral rectangle in cell 1.1 to 1.2.
_____
| |
| |
| |
| |
_____
1.1 1.2
More later,...
What I am getting at is that in a upward slope the right triangle whose tip is 1.44 hinged at the midpoint 1.15 predicts that future point in the derivative as the right triangle hypotenuse.
But the geometry is different for a downward slope function such as 10 -x --> Y. In this case we have the rectangle integral, but instead of hinging up the right triangle to predict the next point of the function graph, we totally remove the right
triangle from the graph and the missing right-triangle is the successor point.
Teaches that derivative predicts next point of function graph--silly Old Math has derivative as tangent to function graph unable to predict. The great power of Calculus is integral is area under function graph thus physics energy, and its prediction
power of the derivative to predict the next future point of function graph thus making the derivative a "law of physics as predictor". Stupid Old Math makes the derivative a tangent line, while New Math makes the derivative the predictor of next point of
function graph. No wonder no-one in Old Math could do a geometry, let alone a valid proof of Fundamental Theorem of Calculus, for no-one in Old Math even had the mind to realize Calculus predicts the future point in the derivative.
TEACHING TRUE MATHEMATICS-- only math textbooks with a valid proof of Fundamental Theorem of Calculus--teaches that derivative predicts next point of function graph--silly Old Math has derivative as tangent to function graph unable to predict. This is
why calculus is so important for physics, like a law of physics-- predicts the future given nearby point, predicts the next point. And of course the integral tells us the energy. Silly stupid Old Math understood the integral as area under the function
graph curve, but were stupid silly as to the understanding of derivative-- predict the next point as seen in this illustration:
From this rectangle of the integral with points A, midpoint then B
______
| |
| |
| |
---------
To this trapezoid with points A, m, B
B
/|
/ |
m /----|
/ |
| |
|____|
The trapezoid roof has to be a straight-line segment (the derivative)
so that it can be hinged at m, and swiveled down to form rectangle for integral.
Or going in reverse. From rectangle, the right triangle predicts the next successor point of function graph curve of B, from that of midpoint m and initial point of function graph A.
AP
Archimedes Plutonium's profile photo
Archimedes Plutonium
1:04 PM (4 hours ago)
to
In the case of a upward slope function, the derivative requires a midpoint in the integral rectangle for which the right triangle is hinged at the midpoint and raised to rest upon the 4 sided trapezoid that the rectangle becomes. Thus the vertex tip of
right triangle predicts the next future point of the function graph by this vertex tip.
However, a different situation arises as the function graph has a downward slope. There is no raising of a right triangle cut-out of the integral rectangle. And there is no need for a midpoint on top wall of the integral rectangle. For a downward slope
Function Graph, we cut-away a right triangle and discard it. Here the vertex tip is below the level of the entering function graph and is predicted by the derivative.
So there are two geometry accounting for the Fundamental Theorem of Calculus proof. There is the accounting of a function graph if the function has a upward slope and there is the accounting if the function graph is a downward slope. Both involve the
Integral as a rectangle in a cell of whatever Grid System one is in. In 10 Grid there are 100 cells along the x-axis, in 100 Grid there are 100^2 cells. If the function is upward slope we need the midpoint of cell and the right triangle is hinged at that
midpoint. If the function is downward slope, the right triangle is shaved off and discarded-- no midpoint needed and the resultant figure could end up being a rectangle becoming a triangle. In the upward slope function graph, the rectangle becomes a
trapezoid, possibly even a triangle.
AP
Archimedes Plutonium's profile photo
Archimedes Plutonium
3:32 PM (2 hours ago)
to
So for an upward slope function, the Proof of Fundamental Theorem of Calculus would have the integral rectangle turned into this.
______
| |
| |
| |
---------
To this trapezoid with points A, m, B
B
/|
/ |
m /----|
/ |
| |
|____|
While for a downward slope function, the Proof of Fundamental Theorem of Calculus would have the integral rectangle turned into this.
______
|....... |
|....... |
|....... |
---------
|\
|...\
|....... |
---------
Where the right-triangle is now swiveled at midpoint but rather where a right triangle is cut-away from the Integral that is a rectangle and that right triangle is then discarded.
Now two of the most interesting and fascinating downward slope functions in 10 Grid of 1st Quadrant Only would be the quarter circle and the tractrix.
Many of us forget that functions are Sequence progressions, starting at 0 and moving through all 100 cells of the 10 Decimal Grid System.
Here, I have in mind for the quarter circle a radius of 10 to be all inclusive of the 10 Grid.
AP
By insisting that the only valid function in the world is a polynomial function, we thus reduce Calculus to the ultra simple task of the Power Rule.
So we have a function of x^3, the derivative by Power Rule is (3)x^2. The integral by Power Rule is (1/4)x^4, and to check to see if integral is correct, we take the derivative of (1/4)x^4 to see if it becomes x^3, and surely it does so.
So what AP teaches math to the world, is that Calculus can be mastered by 13 and 14 year olds. Students just beginning High School.
Impossible in Old Math because Old Math is filled with mistakes and errors and crazy idiotic and stupid math.
In New Math, we clean house. We do not let creeps and kooks fill up math that causes students to have nightmares and nervous breakdowns and vomit before tests.
In New Math, we think only of our young students, we do not think of kooks like Dr.Hales, Dr.Tao, Dr. Wiles trying to achieve fame and fortune at the expense of our young students-- who, all they wanted was to learn the truth of mathematics.
If you run to a teacher of New Math with a function, and that function is not a polynomial, then the teacher is going to tell you "that is not a valid function, and you simply convert it to a polynomial".
In AP math class in 9th grade USA, AP makes students of 13 and 14 year old master Calculus. Master calculus better, far better than 1st year college students in Old Math at any college or university across the globe.
14 year old students in AP math class master calculus and "have fun and joy" in math class.
19 or 20 year olds in colleges and universities go through nightmares, vomiting, and even nervous breakdowns in their learning calculus.
I am not exaggerating here, but obvious observations of education of mathematics.
No-one in math education cares about students in Old Math. No-one has ever Cleaned House of Old Math, but let the rotten fetid Old Math stench increase.
Now I need to add more to the Power Rules of Calculus as we make Polynomials be the only valid functions of mathematics. If you come to math with a function not a polynomial, you are sent home to convert your silly contraption into a polynomial over a
interval in 1st Quadrant Only, a interval of concern.
But in all the years I did calculus, I seem to not have registered in my mind the geometrical significance of the Power Rules. What is the geometry of taking x^2 to the power rule of n(x^n-1) for derivative. Then what is the geometry significance of
taking the integral power rule-- (1/(n+1)) (x^(n+1)).
It seems to me that at one moment in time, that geometry stuck to my mind, but is now elusive, I cannot recall the geometry significance of either Power Rule when played out on x^n.
Cavalieri 1598-1647
So that if we start with a polynomial function such as x^2 -> Y, we instantly know from the power rules that the derivative is 2x and the integral is 1/3x^3.
Derivative Power Rule of a polynomial x^n that the derivative is n(x^n-1).
The Integral Power Rule is sort of the opposite of the derivative rule so for polynomial x^n that the integral is (1/(n+1)) (x^(n+1)).
On Tuesday, September 5, 2023 at 3:00:37 AM UTC-5, Archimedes Plutonium wrote:
Now I need to add more to the Power Rules of Calculus as we make Polynomials be the only valid functions of mathematics. If you come to math with a function not a polynomial, you are sent home to convert your silly contraption into a polynomial over a
interval in 1st Quadrant Only, a interval of concern.
But in all the years I did calculus, I seem to not have registered in my mind the geometrical significance of the Power Rules. What is the geometry of taking x^2 to the power rule of n(x^n-1) for derivative. Then what is the geometry significance of
taking the integral power rule-- (1/(n+1)) (x^(n+1)).
It seems to me that at one moment in time, that geometry stuck to my mind, but is now elusive, I cannot recall the geometry significance of either Power Rule when played out on x^n.
Cavalieri 1598-1647
So that if we start with a polynomial function such as x^2 -> Y, we instantly know from the power rules that the derivative is 2x and the integral is 1/3x^3.
Derivative Power Rule of a polynomial x^n that the derivative is n(x^n-1).
The Integral Power Rule is sort of the opposite of the derivative rule so for polynomial x^n that the integral is (1/(n+1)) (x^(n+1)).
Now I need to include the Cavalieri proof, a geometry proof that rectangles under a function graph such as Y--> x^2 yields the power rule formula (1/(n+1))(x^(n+1)) so for x^2 the integral is (1/3)x^3.
I would think that showing Cavalieri's proof would be standard fare in all 1st year college calculus textbooks. To my surprise, not Stewart, not Apostol, not Fisher& Zieber, not Ellis & Gulick, not Strang, no-one is up to the task of showing how
Cavalieri got that formula from summing rectangles.
Morris Kline in volume 1 "Mathematical Thought" shows a picture.
Stillwell in "Mathematics and its History" shows a picture.
But it must be too difficult for college authors to replicate Cavalieri's proof of approximating rectangles for x^2.
Now if I were back in the days of Cavalieri and tasked to find a formula, I would do rectangles and trial and error. First finding a formula for easy ones such as Y--> x, then Y-->x^2, then a third trial, Y--> 2x to see if the formula is good, sort of a
math induction settling upon (1/(n+1))(x^(n+1)).
But I am very disappointed that none of my college calculus books derives the formula (1/(n+1))(x^(n+1)) via approximation.
There were no standards for math proof in the days of Cavalieri for his genius of deriving the Integral Power rule. Y--> x^n is integral (1/(n+1))(x^(n+1))
So what I am going to do is prove (1/(n+1))(x^(n+1)) in New Math.
I looked through the literature and there was no actual Old Math proof of (1/(n+1))(x^(n+1))
This is worthy of a whole entire new book of itself.
And the beauty is that it is a Mathematical Induction proof.
And the beauty also is that functions are chains of straightline connections from one point to the next in Discrete Geometry.
That means we no longer approximate the integral but actually derive the Integral from a Right Trapezoid whose area is 1/2(base_1 + base_2)(height).
We see that in a function such as 3x becomes integral (1/2)(3)x^2 due to that right-trapezoid area.
The right-trapezoid is such that its base_1 and base_2 are the Y points for cells of calculus in Decimal Grid Systems.
Trouble in Old Math is when the "so called historian" reads a passage in old works, they become overgenerous in crediting a proof when none really existed -- Fermat, Cavalieri. And this is the reason that no-one in modern times who wrote a Calculus
textbook features the Cavalieri Integral Power Rule, because there never was a proof, .... until now... a Mathematical Induction proof.
AP
None of this is a proof of Cavalieri's integral power rule formula. Because Geometry is discrete and all curves in geometry are chains of straightline segments. The Internet boasts of some modern recent proofs of Cavalieri, but I suspect all those are
bogus claims, being victims of computer graphics and no honest down to earth proof at all. I myself was a victim of computer graphics, for a computer can really spit out any image you ask it to spit out, such as hexagon tiling of sphere surface.
--- quoting Wikipedia ---
The modern proof is to use an antiderivative: the derivative of xn is shown to be nxn−1 – for non-negative integers. This is shown from the binomial formula and the definition of the derivative – and thus by the fundamental theorem of calculus the
antiderivative is the integral. This method fails for
∫1/x dx
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