Any proof for the definition of the definite integral

[mikie1]

Hello all, new poster here.

When learning about the definition of the definite integral, a few books that I have read through first define anitdifferentiation, then explains some Riemann sums and then

POOF

\int_{a}^{b} f(x) dx = \lim_{||\Delta||\rightarrow\0}\sum_{i=1}^n f(\xi_i) \Delta_i x

So is there a proof for this or do I just got to accept it? Please help with the integral definition.

 

[HallsofIvy]

Depends upon exactly what you mean. One doesn't have to "prove" definitions: they are given. Of course, one might want to prove that the definition actually gives you something worth while! In this case, it doesn't, necessarily: there exist many functions that are N0T "integrable" and for which the definition of the definite integral doesn't make sense.

Probably you will have to get a lot of calculus under your belt and then take either an "Advanced Calculus" or a "Mathematical Analysis" course to see a real proof that continuous functions are always integrable. (But there exist non-continous functions that are also integrable. I didn't see a theorem stating exactly what functions are integrable until I was in graduate school.)

Roughly speaking- it goes like this: divide your x-axis (from a to b) into segments and, on each segment choose x^* so that f(x^*) is the largest possible on that segment. Turn around and choose x_* so that f(x_*) is the smallest possible on that segment. If x is any point on the segment, then by definition f(x_*)<= f(x)<= f(x^*) so,taking the length of the segment to be &Delta; x, f(x_*)&Delta; x<= f(x)&Delta; x<= f(x^*&Delta; x and then &Sigma;f(x_*)&Delta; x<= &Sigma;f(x)&Delta; x<= &Sigma;f(x^*&Delta; x .

IF f is continuous then you can show that, in the limit, x_* goes to x^* so the two end sums go to the same thing. Since the sum for ANY choice of x is "trapped" between them, it must also give the same result: the definite integral.

By the way, if you think of this in terms of "area under the curve", notice that x^* always gives rectangles that include the area while x_* always gives rectangles that are included in the area so that the "area" always lies between the two. Of course, the area is a constant so if the two limits are the same then that common value must BE the area. That is, one can show that the definite integral gives the area under the curve without having to give a precise definition of "area"!

The proof that one can get the definite integral by evaluating the anti-derivative at the endpoints is a different matter and is quite often given in calculus books.

 

[Michael D. Sewell]

The whole is equal to the sum of it's parts.

 

[mikie1]

The whole is equal to the sum of it's parts.

And that makes perfect sense for the Riemann sums, but since when did taking a function and increasing its exponent value by one and divide that by the value in the final exponent equate to the area under the curve (simple case). That is where there is a leap of faith for me.

With Simpson's rule, one uses some algebra and calculus to prove that the area under the curve is

1/3 h (y0 + 4y1 + y2)

 

[Severian596]

This page seems to provide some interesting historical information about the derivative and finally the integral. It's a bit heavy depending on your level of experience, but very wordy, so stick with it until the end.

http://www.math.wpi.edu/IQP/BVCalcHist/calc2.html

Section 2.5 The Ellusive Inverses – the Integral and Differential is what caught my eye...seems to target your curiosity.

 

[Michael D. Sewell]

And that makes perfect sense for the Riemann sums, but since when did taking a function and increasing its exponent value by one and divide that by the value in the final exponent equate to the area under the curve (simple case). That is where there is a leap of faith for me.

I'll leave it to someone more qualified then I am to give you rigorous proof.

Take a right triangle with two legs of 5 inches each. Take the length of one side(5 inches) and " increase it's exponent by one and divide that by the value in the final exponent" to find it's area. Now stop and look at what you have done, and spend a little time thinking about it. Is this a big leap of faith?
Would you rather use your Riemann sum here?

If you are talking about a position function, there are two ways to arrive at the total distance traveled:
1.) The summation of an infinite number of infinitesimal distances.
2.) subtracting the intial position from the final position.

They both result in the same answer.

The Riemann sum with yield the same result as the definite integral, if it is carried to completion(When delta x = dx). This will take you a long, long, time...

-Mike

 

[HallsofIvy]

With Simpson's rule, one uses some algebra and calculus to prove that the area under the curve is

1/3 h (y0 + 4y1 + y2)

If this is what you really meant to say then you are way off. You can't "prove" that because, in general, it isn't true. Except for curves of polynomials of degree 2 or less, Simpson's rule gives only an approximation to the area.

The idea that the area under the curve is the inverse of finding the derivative (which is what I guess you mean "increase it's exponent by one and divide that by the value in the final exponent") is not a leap of faith- the proof is given in every calculus book- it's a fairly straightforward application of the mean value theorem.

I wondered for a moment what "increase it's exponent by one and divide that by the value in the final exponent" had to do with triangles- it's interesting that it DOES work in the case you give! If you have right triangle with legs of length 5, then I guess the "exponent" of the length of one side is 1: increasing it by 1 means you have 5² and then you divide by 2: yes, by Jove, the area is (1/2)(5²)!

 

[NateTG]

You mean the fundamental theorem of calculus.

Here's an informal version:

Let's say that we have a function F(x) with derivative f(x) on [a,b].

Now, if we partitiion [a,b] with x_i so that a=x_0&lt;x_1&lt;x_2...&lt;x_n=b then F(x_1)-F(x_0) \approx (x_1-x_0) f(x_0)

Now, the left sum of f(x) on the same partition is:
\sum_{i=0}^{n-1}(x_{i+1}-x_i)f(x_i) \approx \sum_{i=0}^{n-1} F(x_{i+1})-F(x_i)

But it's easy to see that the sum on the right hand side collapses into:
F(x_n)-F(x_0) which is the result that you would expect from the fundamental theorem of calculus.

There's a bit of work involved making this formal, and some more work showing that the limit exists, but this should help a bit.

 

[mikie1]

OK, a bit simplified, but here is an example where using a couple of different numbers can be misleading.

0 + 0 = 0

0 x 0 = 0

2 + 2 = 4

2 x 2 = 4

[sarcasm]I have just shown per example that addition and multiplication are the same operators. [/sarcasm] I believe that we all know that addition and multiplication are different operators, but some examples can lead one to believe that.

There are many more complex funtion(s) that have ugly solution(s) after integration. It still seems like pulling a rabbit out of a hat when saying the integral is equal to the area under the curve. Riemann sums is the only way to back it up.

As for the Simpson's rule, I did overstate. I was trying to state that the area under the curve for a parabola is equal to

1/3 h (y0 + 4y1 + y2) and can be algebraically proven.

 

[Michael D. Sewell]

I wondered for a moment what "increase it's exponent by one and divide that by the value in the final exponent" had to do with triangles- it's interesting that it DOES work in the case you give! If you have right triangle with legs of length 5, then I guess the "exponent" of the length of one side is 1: increasing it by 1 means you have 5² and then you divide by 2: yes, by Jove, the area is (1/2)(5²)!

I like that one because it kind of sneaks up on you.

Triangle:
If f(x) = x
then F(x) = 1/2x^2

Pyramid:
if f(x) = x^2
then F(x) = 1/3x^3

mikie1,
As you work with calculus over a period of time, these things get much easier to visualize.
-Mike

 

[Michael D. Sewell]

There are many more complex funtion(s) that have ugly solution(s) after integration. It still seems like pulling a rabbit out of a hat...

mikie1,
I hope your attitude about calculus changes. When you say that calculus is in any way "ugly", it sounds like you're trying to fight with it. Calculus is truly beautiful. I hope that you see this soon, it will make calculus much easier and more enjoyable for you.

My little parlor trick, "or pulling a rabbit out of a hat" as you put it, shows you that you have already been using calculus for years, and you didn't even know it. Take the formula for the circumference of a circle(2 pi r) and integrate it. You now have the formula for the area of a circle(pi r^2).

Take the surface area of a sphere(4 pi r^2) integrate that and you have the formula for the volume of a sphere (4/3 pi r^3). Does any of this look "ugly"?

I think that pulling a rabbit out of a hat is a wonderful solution! The fact that you can see calculus in this way is a great start! -Mike

 

[HallsofIvy]

Now, he didn't actually say that calculus was "ugly"- he said that integrals have "ugly" solutions, by which he seems to mean "not trivial".

This is generally true of "inverse" problems: If you are given some complicated polynomial, f(x), given a and asked to find y= f(a), that's straightforward calculation. If you are asked to find x such that f(x)= a, that's an extremely difficult problem.

The derivative is defined through a simple formula. The "anti-derivative" is defined as its inverse. That's why anti-derivatives tend to be complicated.

Once again, the "fundamental theorem of calculus"- that the definite integral, defined as "area", can be derived through the anti-derivative, is NOT "pulled out of a hat" but has a simple proof given in any calculus book.

 

[mikie1]

Thanks for all the replies!

A little bit about me. I'm 35, took some calc in high school (1986) and graduated nontraditionally in 1995 with a BS in electrical engineering. I have some extra time on my hands and I'd like to clear up some things from my educational past that seemed vague to me or the prof didn't have a good explanation.

One was the Simpson's rule. In a problem solving / computer apps class one of the profs said that with Simpson's rule, we are estimating a curve with parabolas and the coefficients of the parabola must be determined in order to calculate the area under the curve. This is untrue since Simpson's rule just uses three points and a formula. Another one is paint cans that can't be painted.

HallsofIvy - Thanks, I do find that calc can be pretty kewl and U R exactly right on my interpretation of "ugly". I need to be a little more clear since I'm dealing with "purest" on this board.

I'll read through some of the links to see if I can "see the light".

 

[HallsofIvy]

"In a problem solving / computer apps class one of the profs said that with Simpson's rule, we are estimating a curve with parabolas and the coefficients of the parabola must be determined in order to calculate the area under the curve. This is untrue since Simpson's rule just uses three points and a formula."

No, believe it or not, what your professor told you is not untrue. His/her point was that the "formula" you are referring to is derived by approximating the curve by the parabola that goes through those three points.

What you are saying could be interpreted as "No, it's untrue that I have to learn anything- I just have to memorize formulas."

 

[nermeen_sadaam]

hi.. I'm facing a trouble in finding a proof for the derivation of the volume of a sphere, a cylinder, and a cone using integration.. so please provide me with the solution for this assignment that i have & i can't answer.. with my appreciation & thanks a lot for your support
nermeena

 

[vadik]

Let height of the cylinder to be L and the area of the bottom A. Then the volume will be --> \lim_{||\Delta L||\rightarrow\0}\sum_{i=1}^n A \Delta L = \int_{0}^{L} A dL = AL.

 

[HallsofIvy]

hi.. I'm facing a trouble in finding a proof for the derivation of the volume of a sphere, a cylinder, and a cone using integration.. so please provide me with the solution for this assignment that i have & i can't answer.. with my appreciation & thanks a lot for your support
nermeena

I. You are trying to "hijack" a thread on a completely different subject. If you are not responding to something said in this thread start a new one.

II. Since you say this is an "assignment", it should be in the "homework" section and you should show what you have tried. One reason for that is to let us know what kind of help you need. I can think of several ways of doing these but I don't know what methods you already know and are allowed to use.

 

[mikie1]

"In a problem solving / computer apps class one of the profs said that with Simpson's rule, we are estimating a curve with parabolas and the coefficients of the parabola must be determined in order to calculate the area under the curve. This is untrue since Simpson's rule just uses three points and a formula."

No, believe it or not, what your professor told you is not untrue. His/her point was that the "formula" you are referring to is derived by approximating the curve by the parabola that goes through those three points.

What you are saying could be interpreted as "No, it's untrue that I have to learn anything- I just have to memorize formulas."

You are reading into that way too much!

 

[ydnef]

Maybe this helps

To Mike1 original question:

What I always have in mind, when I think of definite integrals is this (though someone already told me that I am wrong, but I don't care)

I define the integral sign to mean

b/
|f(x)=f(a)+f(a+dx)+f(a+2dx)+...+f(b-2dx)+f(b-dx)+f(b)
a/

in direct analogy with ordinary the summation sign.

(INTERMEZZO (skip it if you like):
The ordinary summation sign uses a discreet index n, with n=0,1,2,...
The integral sign uses an index x, which is not a discreet index, but a continuous index. How do we generalize from a discreet index to a continuous index? Well the best way we can do is to invent a "dx" which is supersmall by definition, and this way we can interpret my previous definition of the integral sign almost "as if" it "slides" through the continuum. When you think about it, this is kind of weird, didn't Cantor proof that there we more natural numbers than real numbers? Anyway, I am not smart enough for that stuff.END OF INTERMEZZO)

The "dx" here is Leibniz's and Newton's infinitesimal.

Now, you may think that this is useless, because you never see

b/
|f(x)
a/

without the "dx", that is, you always see

b/
|f(x)dx
a/

Okay, let's do that then.

b/
|f(x)dx=f(a)dx+f(a+dx)dx+f(a+2dx)dx+...+f(b-2dx)dx+f(b-dx)dx+f(b)dx
a/

An infinite number of terms to add, it looks hopeless. But hey, Leibniz found a trick. What if somehow we know that for some F(x) we know that

dF(x)/dx=(F(x+dx)-F(x))/dx=f(x)

You might wonder how that could possibly simplify things, but I am going to do it anyway: replacing f(x) by (F(x+dx)-F(x))/dx we get

b/
|f(x)dx=
a/

[(F(a+dx)-F(a))/dx]dx+[(F(a+2dx)-F(a+dx))/dx]dx+[(F(a+3dx)-F(a+2dx))/dx
dx.....[(F(b-dx)-F(b-2dx))/dx]dx+[(F(b)-F(b-dx))/dx]dx+[(F(b+dx)-F(b))/dx]dx

Now because the "dx" in

b/
|f(x)dx
a/

is chosen by me (and if you are smart you choose the same) to be exactly the same as the "dx" in

(F(x+dx)-F(x))/dx

the two "dx's" cancel each other out, so that

b/
|f(x)dx=
a/

F(a+dx)-F(a)+F(a+2dx)-F(a+dx)+F(a+3dx)-F(a+2dx)...F(b-dx)-F(b-2dx)+
F(b)-F(b-dx)+F(b+dx)-F(b)

The F(a+dx) term at the beginning can be found again three positions further up, only now with a minus sign, so they add up to zero. So it is with most of the terms, and as you can find out for yourselves, only two terms are left

b/
|f(x)dx=F(b+dx)-F(a)
a/

A miracle has happened right in front of your eyes! No need to add together an infinite number of terms, all we have to do is know the anti-derative of f(x).

You might be bothered by the fact that instead of F(b)-F(a) my derivation actually gives F(b+dx)-F(a), but the difference is only infinitesimal.

I warned you, somebody "authorotive" already told me that my derivation is wrong, somehow, in a way I do not understand, it has to to with the mathematical rigidity of me using the infinitesimal "dx", but I do know that the great great Riemann has used his brains to crack this one, so yeah, probably I am oversimplifying things to much, like always, but when I am going to have kids someday, this is the way I am going to explain it to him.

So what do you think of my definition? I have a lot more stupid stuff. For instance, I define the dirac-delta function to be

dirac(x-a)= 1/dx if x=a
0 if x=not(a)

 

[matt grime]

As you got Cantor's cardinals the wrong way round, can I beseech you on behalf of the mathematical community not to pass on this knowledge for so many reasons (misuse of infinitesimals, presumption that adding up an uncountable number of non-zero objects is remotely rigorous, writing a 'function' as equal to the reciprocal of a 1-form). Whatever the suggestive nature of things might be (and notations have evolved to be deliberately suggestive) doesn't justify this level of abuse. If you want proof the why you shouldn't do this as it leads you to get the wrong answers consider:

suppose F(x,y,z)=0 defines implictly x,y, and z as a funtion of each other then what is:

\frac{\partial x}{\partial y}\frac{\partial y}{\partial z}\frac{\partial z}{\partial x}

naively, according to the above, it probably is 1, when it isn't, it's -1. Try working out why

\frac{d^2y}{dx^2} \neq \frac{1}{\frac{d^2x}{dy^2}}

as well.

 

[ydnef]

Help me get my undestanding of infinitesimals right?

To M. Grime:

Yes, having switched the real numbers with the natural numbers is pretty embarrassing.

I have a question. You say that my understanding of the infinitesimal is totally wrong. Maybe you can help me get it right. Let me explain my thoughts on the subject and then maybe you could correct me, or anybody.

Let

Ndx=h

with N some finite integer, dx some finite real number (it is not a infinitesimal yet but is will become one later) and h also some finite real number.
Now let's shrink dx, but I want to keep h a constant.This means that N must become larger, and not just that, I want to keep N an integer too, which forms a restriction on what values dx can take. So h is the only really arbitrary number here, as long as it is finite.

Now let us assume that my shrinking powers are such that I can shrink dx so much that it becomes appropiate to call dx a infinitesimal. What is the value of N to satisfy the equation Ndx=h? Well, I think that N must equal h/dx. Is this absurd, N=h/dx?

Now what I think is that even with dx being a infinitesimal, one will not be able to go through ALL of the real numbers between the interval 0 and h. I do NOT think that dx means the difference between nearest-neighbours on the real number system continuum. The infinitesimal is just a limit. So to me, your earlier objection that I am trying to count an uncountable infinity does not apply. By using dx I was not using all the real numbers of the uncountable infinity, I only was merely using a countable subset of that uncountable infinity, which is ALMOST like going through all the real numbers.

But I am still worried about N=h/dx. I have "discovered" a proof of the natural number e, which I feel is very elegant, but it merrily uses expressions like N=h/dx. This proof of e, which I regard as my greatest personal triumph in trying to understand infinitesimals, will collapse if you can explain to me why N=h/dx is such nonsense.

Did I mention that this is the N I am using in my definition of the integral? Let's rewrite Ndx= h to (h/dx)dx=h. Here the dx in the denominator cancels the dx in the numerator. I CHOOSE both of the dx's to be exactly the same, because I WANT h on the right handside of the equation.

Now you also have a argument involving partial deratives. I am still thinking about that. But "partial" deratives may be beside the point. Because when you write (dx/dy) you implicitly also mean to keep to keep the variable z a constant, and when you write (dz/dx) you implicitly mean to keep the variable y a constant. That may be why the dx in (dx/dy) does not simply cancel the dx in (dz/dx), they are both used in different contexts, only the notation does not reflect that. Well. Maybe. Like I said, I am still thinking about that.

I know very well that most of you (probably all of you) do not agree with me. That is why it is so important that, despite my limited brainpower, one of you with your scary monstertalent for maths tries to help me get it right. (But don't use words like 1-form, I have no idea what you mean, and besides, as you may have noticed, I am not a real mathematician)

ydnef

 

[matt grime]

If all you are doing is saying that 'infinitesimal' means some suitably small real number, then you're doing a numerical method of integration, and you're using something akin to the uniform subdivision of the interval (fixing N to be a natrual number). Moreover, when there is an anti-derivative it drops out quite nicely. That is nothing special, and as almost every function doesn't posses an antiderivative it's of only partial importance, but good to know anyway at high school. Thus you've just proved, albeit without any rigour, the fundamental theorem of calculus

There is a difference between the infinitesimal idea of approximation, where one ought to write \deltax, in calculus and the rigorous manipulation of dx which is not even a real number. So what you write whilst being suggesive is not correct. So I suppose the first thing to say to you is dx is not a real number. The second is that you're just justifying the idea that integration is the reverse of differentiation and can be done to a reasonable numerical degree. Sort of moving the furntiure around. And as long as you stop using dx as a number when it isnt' one you're fine. If you treat them like numbers you will eventually get the wrong answers becuase things happen that are counter intuitive.

 

[ydnef]

You lost me when you said that dx is not a real number. Let me try to explain why I think it is real number by using an example.

Does there exist a f(x) such that (df(x)/dx)=f(x)? Well suppose there is? (We already know there is but let's pretend we do not know yet)

Written full out this equation is

(f(x+dx)-f(x))/dx=f(x)

For sure the dx here represents an infinitesimal, there can be no doubt at all what dx is here. We can rewrite the equation to

f(x+dx)=(1+dx)f(x)

According to this shifting by dx on the x-line just means multiplying f(x) by (1+dx). So

f(x+2dx)=((1+dx)^2)f(x)=...=(1+2dx+dx^2)f(x)

f(x+3dx)=((1+dx)^3)f(x)=...=(1+3dx+3dx^2+dx^3)f(x)

f(x+4dx)=((1+dx)^4)f(x)=...=(1+4dx+6dx^2+4dx^3+dx^4)f(x)

What do we see appearing as the coefficients? Pascal's triangle
1
11
121
1331
14641
...

We can replace the expressions by the binomialcoefficient (I think it is calles that) and make a general formula for

f(x+Ndx)=...(some formula which contains the binomialcoefficient)

Now I take Ndx=h in the way like I explained before and ...(I shortcut the thing now, the internetcafe where I am in is closing)...and in the end I get the series

1+1/2!+1/3!...

the series one generally refers to as e.
So what I did is, I took the infinitesimal dx and treated it like a real number and voila, I get e. So why didn't things go wrong then.

I'll check the net again tomorrow, hopefully you're not tired of me.

ydnef

 

[matt grime]

All you're doing is estimating the integral with a numerical approximation, which if there is an antiderivative yields the obvious answer.

dx is not a real number, it is not a number at all, if you're saying it is then you don't mean dx, you mean \delta x which is tiny. Look up these things. You are just doing the numerical integral with equal intervals of length dx (when you shouldn't be) and then saying as dx goes to zero you get the anti-derivative. And? That is rather obvious and taught to every undergrad, with the proper notation.

Treating it as a proper number can reveal some obvious answers, and the way you generate e is exactly the way we were taught as callow 16 year olds when attempting to find a function whose derivative is itself.

So, in some sense, congratulations, you've got what we already know and teach. It's just that you're using the wrong names for things. Just use a delta x, do the manupulation, let it tend to zero and voila the answer, just as is supposed to be.

edit actually we do know when diff eqns have solutions and when they are unique.

 

[ydnef]

Okay, for me this could be the last time for me because we can't seem to convince each other.

What I think you are saying is this:
Manipulating dx is a tricky business. Sometimes they cancel each other out and we are fine. But there many more instances when they do not simply cancel each other out and you have to be careful.

Maybe you mean when you say that dx is not a number, you mean that dx does not represent the same number all the of time, it depends on the context, and in that sense it is just a symbol.

Let me elaborate. When you say that I have to be careful cancelling the one infinitesimal dx by the other infinitesimal dx, you mean that in the same way as you would say that one has to be careful cancelling the one infinity 8 by the other infinity 8 (My keyboard doesn't have a the correct symbol for infinity, so I'll make do with the 8, which is the same only turned 90 degrees)
Infinities come in many forms, they can go to infinity logarithmic, quadratic or linear,etc. The result of the division infinity/infinty=8/8 depends totally upon what the symbols actually represent. Naively by just looking at the symbols, you would say that they cancel and the result of the division is one, because they look the same. But only when you know for a fact that the two dx's represent the same thing then you can cancel them. If you do not know that then be careful, because the symbols do not keep track of what type of infinity is really meant.

And so it is with the dx's. The dx's are just symbols. Off course one has to be careful in cancelling the one dx by the other dx, because they might not represent the same thing, they are just symbols, and you have to check what they represent. But in my case, when I define the integral, I happen to know for a fact that they are the same. What I am going to show you now may not be rigourous enough for you and I hope you won't have an heartattack:

Let's define the integral like I wanted to:

b/
|f(x)dx=f(a)dx+f(a+dx)dx+f(a+2dx)dx+...+f(b-2dx)dx+f(b-dx)dx+f(b)dx
a/

How do I know that the dx inside f(a+dx) is the same dx as which I multiply it with? The answer: Because I want to give the integral an interpretation, namely I want it to mean the following

b/
|f(x)dx="area underneath the curve f(x)"
a/

(for simplicity take f(x)>0)

Take time to convince yourself that the integral can only take on that interpretation if and only if the dx inside f(a+dx) is the same as the dx you multiply it with.
Now let's substitute f(x)=dF(x)/dx into f(a+dx) (wether that is possible is like saying wether a function is integrable or not).
We get

[f(a+2dx)-f(a+dx)]/dx

This says that the dx in the denominator is the same as the dx in f(a+dx), which as I showed is the same as the dx we multiply it with, and thus the two dx's cancel out. That is

([f(a+2dx)-f(a+dx)]/dx)dx=f(a+2dx)-f(a+dx)

What I think I have shown is, that in order to get the interpretation right, namely that of "area under a curve", I have to choose all the dx's, in this special case, to be the same. My desire to get a certain interpretation for the integral forces me to make all the infinitesimals to be the same thing. If I do not, the integral could not be interpreted like that. In this special case I do not need to find out what type of infinitesimals the dx's actually represent because I discovered that to get the interpretation right they must be the same and they cancel out and we don't have to worry about them.
This does NOT AT ALL quench your thirst for rigour, does it?

Why is the fact that I represent the integral as a discreet sum so disturbing to you (not only you I suppose). Maybe it is because you think that all a discreet sum can lead to is an approximation and could never define the integral perfectly. But you forget that I am using the infinitesimal dx. It does not matter that the sum is discreet, by using the infinitesimal dx we ensure that we do not get an approximation but an exact answer. That is the power of our beloved infinitesimal. At first sight you would not think that such a small thing could be such a powerful tool. Numerical integration does not make use of this wonderful tool. It is in this sense that I can say that what I do in doing integrals is not like numerical integration at all, because numerical integration does not make use of that one special ingredient.

So to me dx as a symbol represents a real number. I know dx/dx does not automatically mean 1, because the dx in the denominator might mean
(x(z+dz)-x(z)), whereas the other dx might mean (x(y+dy)-x(y)), one has to check, but in my specific case, there are no worries. But the fact that one has to be extremely careful in other circumstances does not mean that dx is not an element of the real numbers. It just means that you have to careful. Well, to me at least.

But hey, I do appreciate it that you are trying to teach me something. Nobody else seems to care that I am wrong.

 

[matt grime]

"Now let's substitute f(x)=dF(x)/dx into f(a+dx) (wether that is possible is like saying wether a function is integrable or not)"

No, it is saying that there is an antiderivative. There are functions that do not have one (in a nice sense) but are still integrable.


You are just doing a numerical integration with dx instead of [tex[\delta x[/tex] in the particular case where the integrand has an anti-derivative and all the subintervals have the same length.

The integral is not the same as the sum, and your assertion that

df/dx = (f(x+dx)-f(x))/dx

is not correct. Try using deltas and taking the limit at the right time and you'll see why the integral is what you think it ought to be for this particular kind of integrand.


I'd rather not get into a debate about what dx is. But you cannot claim that there are N dx's in the interval and that dx is an infinitesimal, you're mixing and not matching what you mean so that your method is forced to be correct. If you stuck with the proper (conventional) definitions of these things then you woldn't have a problem.

 

[Hurkyl]

What you're doing is certainly the kind of intuition that went into the definitions of these things; the probelm is that it's fairly unsuitable to use in practice. One of the major advances in the past hundred or so years was the development of a rigorous treatment of such concepts so that we didn't have to rely on brash, conflicting, and erroneous intuitive understandings to guide us.


First off, just saying "dx is an infinitessimal" is not enough. You've implicitly assumed you can do things like add dx to a real number and multiply dx by a real number.

And there's the very difficult logical hurdle about just what the heck "f(a+dx)" can mean when f is a function whose domain is the real numbers, but a+dx is not a real number. (It is a real number plus an infinitessimal)

After that, there's another difficult hurdle about what "f(a)dx+f(a+dx)dx+f(a+2dx)dx+...+f(b-2dx)dx+f(b-dx)dx+f(b)dx" could possibly mean.

Addition only adds two things. Repeated application of addition can only add a finite number of things.

We have to use a trick (limits of partial sums) in order to add up a countably infinite number of things, and this addition isn't commutative (sometimes you can rearrange terms and get a different sum) nor is it associative (sometimes you can regroup terms and get a different sum)... However, if dx is infinitessimal, this sum has uncountably many terms, and thus we can't even apply this trick.



Now, there are ways to solve these problems, which are used in nonstandard analysis... but while nonstandard analysis has infinitessimals, dx is not one; it still remains a mere symbol. Formulas similar to the ones you write are used, but there is more detail necessary to do it correctly.

 

[ydnef]

Yes, that things may not commute, is a good argument, I have seen that somewhere before and you are right. But the thing about the number of infinitesimals inside a finite interval being uncountable, well, for now I do not get that. And yeah, for instance, the dirac-delta function does not work well with the fundamental theorem of calculus, I know such things exist.

One more thing. How about my method being a pedagogical device, would that work? The way you are pressing on and on about rigour will not help the first time student at all. Maybe it is not the most powerful definition of the integral, in the sense that it not the most general, and my God, things can really get generalized to the most abstract, as we know!

Anyway, I know that when you go to higher maths things get a little bit different, but I somehow cannot accept that probably in my lifetime I will not truly ''get it'', and that all the teachers I encounter tell me that things are more complicated, which I can accept, but when I ask for an explanation, they give me vague sentences and insult my intelligence, and say...well I don't understand what they say, probably because they themselves do not know the precise definition. That's true right? Most people (I'd say more than 99.9 percent) who are excellent in doing integrals in the end do not really know what are really doing because that thing is way up there in what you call non-standard analysis. All they know is that things are more complicated than that.

Ydnef

 

[matt grime]

As we've said, what you're getting at is the idea behind integration, but that you aren't using infinitesimals in the correct manner, never mind just rigorously. An infinitesimal isn't a real number (with the exception of 0). As hurkylu points out the dx there isn't an infinitesimal. It is a symbol, it is not a real number.

You can't have something being the definition of the integral and not being the integral at the same time. Basically you're taking the analyitc idea of lots of intervals of length \delta x and omitting the explicit mention of taking the limit as it tends to zero. That is all. You can't omit steps just because you don't like them when they are neceessary.

So, once more, the problem is that you are writing dx as if it were a real number, and it isn't.

WE aren't doing non-standard analysis to do integration. The integral has been defined many times in this thread (Riemann integral) as ther area under the curve. What's so hard about that?


Edit: here's the "proper" version of what you're doing.

Let F be a differentiable function, so that f(x)=F'(x), then diffble means that F(x+\delta ) = F(x)+\delta F&#039;(x) + \delta o_x(\delta) where o_x is some function that tends to zero as delta tends to zero.

Then using the division of [0,1] say into n equal intervals of length 1/n :=delta

\sum_r \delta f(r/n) = F(1)-F(0) +\sum_r \delta o_{r/n}(\delta) noting all the cancellations we can do; the key is that we can then take max of the o's add em all up and get something that tends to zero as n tends to infinity.

Thus as a partition exists and all that with the upper and lower sums etc, it is Riemann integrable with integral F(1)-F(0)

Really we're just proving the FTC.

 

[jdavel]

mikie1,

The answer to your question may be somewhere in this thread.

If not:

The definition of the definite integral as the limit of an infinite sum, is just that, a definition. So it's true by definition. And yes, you have to accept it.

But I suspect that what you really meant to ask is whether this has been proven: Given a function, continuous on a closed interval [a,b], the difference between the values of the anti-derivative at a and b is the same as the definite integral of the function on the interval [a,b]. This is the fundamental theorem of calculus, and yes, like all theorems, it's been proven. If you didn't see the proof when you first learned integral calculus, either you were asleep or your teacher should be fired!

 

[Hurkyl]

The tone of my responses generally reflects the tone I perceive in a thread, which may or may not accurately reflect the intended tone of the posters.

But the thing about the number of infinitesimals inside a finite interval being uncountable, well, for now I do not get that.

The basic idea is that for each point x in the interval, there will be at least one term infinitely close to x. Since there are uncountably many real numbers in any interval, your sum must have uncountably many terms.

The way you are pressing on and on about rigour will not help the first time student at all.

Well, one should know the right way to do things; while it is important to get an intuitive grasp of the concepts in question, it is also important to learn how to translate intuition into rigor, and to learn how to apply rigor when intuition fails, or worse, misleads.

"A little learning is a dangerous thing"

When learning any concept (not just in mathematics), it is just as important to learn how things go wrong as it is to learn how things go right.


Anyways, the main point I'm trying to make is that your presentation of the integral is not what it "is"; it's what it "essentially is".

 

[ydnef]

On the one hand I am tired of having to sit in this stupid internetcafe, but on the other hand, I cannot let it go and your remark about non-associativity got me thinking. I think I might now understand better what one means when one says that dx is not a real number.

In some maths-book I once saw this sum where, if one would put the different terms in a different order, the outcome would also be different. But there are as far as I remember at least two requirements needed for that to happen:

1)The sum needs to add up an infinite number of terms
2)The terms cannot all have the same sign, some of them must be positive, some of them negative

Now, the terms inside this specific summation (sorry I can't produce an example but I think you know what I am talking about) are all real numbers. And real numbers, by definition, all associate, meaning a+b=b+a. But it turns out that even real numbers do not associate when you have an infinite number of them. BUT ONLY IN THE CONTEXT OF AN INFINITE NUMBER OF TERMS. One could be difficult and demand that real numbers always associate and therefore the terms in the summation are not real numbers. I would never use those words to describe this phenomenon because it is confusing me. I would just keep calling those terms real numbers.

Now infinitesimals used in integrals are always in the context of ''infinite number of terms'' because you need an infinite number of infinitesimals to do integrals. Integrals have that context built into them. So because of that infinity-game you play, you cannot guarantee associativity. One could say that the infinitesimals are not behaving like real numbers, or put it even more strongly and say that dx's are not real numbers, which is just a way of giving words to a mathematical idea. To me, one expresses his or herself better when on says that, YES, the dx's are real numbers, BUT dx's used inside integrals are always in the context of infinite number of terms, SO you know what holds for real numbers when one has an infinite number of them, namely the breakdown of associativity, also must hold for integrals infinitesimals in the context of integrals.

If one insists that real numbers always associate, then the integral is a case where one cannot call the the dx's real numbers.

My definition, which sort of gets the spirit of integration, says that

b/
|f(x)dx=F(a+dx)-F(a)+F(a+2dx)-F(a+dx)+...-...+...-...F(b+dx)-F(b)
a/

We see that:
1)There are an infinite number of terms to be added
2)The terms do not all have the same terms
For me to get to the statement

b/
|f(x)dx equals F(b+dx)-F(a)
a/

I would have to reshuffle the terms before I can cancel most of them with each other. But that is the thing, I am not allowed to ''just'' reshuffle them.
Anyways, am I right that saying ''dx's are not real numbers'' is bad terminology? Surely one must acknowledge that bad terminology and lousy notation can hold back advances in understanding.

But if what I am saying is true, then the fact that (dx/dy)(dy/dz)(dz/dx) equals -1 and not +1 for F(x,y,z)=0, cannot be attributed to the other fact that, in your terminology, dx's are not real numbers, because in differentiation, you do not need an infinite number of infinitesimals to get results, you just need a few of those dx's.
So the flipping of the sign must be because of something else. Could the reason perhaps be what I remarked earlier about partial differentiation, that (dx/dy) implicitly keeps z constant and (dz/dy) keeps implicitly keeps x constant, and because of the two different situations when performing the differentiations one cannot simply cancel the two dx's?
Or did I not understand all the reasons for people to call dx not a real number and are those reasons that I do not yet comprehend responsible for this peculiar flipping of the sign?

By the way, what were Newton and Leibniz actually thinking when they did not know of non-standard analysis?

Do not interpret any undertone in my writing as being not well-wishing to you or anybody. Read it as me hating my own failure to understand. Any replies I am grateful for.

 

[matt grime]

The reason you shouldn't call dx, infinitesimals or whatver, as real numbers is because they aren't real numbers. The same way that 2x3 matrices aren't real numbers, the same way that the set of funtions on the unit disc aren't real numbers. They aren't. Nothing to do with the sum being an an infinite one. And you mean, seeing as you think notation is important, that addition the real numbers is associative, not that the "real numbers associate" (back-formed verbs are annoying at the best of times).

The symbol int F(x)dx does not mean add up anything with dxs in it at all, it denotes the integral of F, in whatever sense of integral we are using, usually riemann, where it involves upper and lower sums, partitions, and such.

Have you noticed how lots of your statements begin 'if what i think were true...' well, as a general rule it isn't true, now matter how nice it seems in your opinion. The dxs in ordinary analysis denote a limiting process from some\delta x as it tends to zero. As we;ve said, you've got what the concept of integration is getting at, but it isn't how the integral really works.

 

[Hurkyl]

The easiest example of nonassociativity is:

(1 + -1) + (1 + -1) + ... = 0
1 + (-1 + 1) + (-1 + 1) + ... = 1



Anyways, the trick to dealing analytically with infinity and infinitessimals is to use limits; they're what makes (standard) analysis tick. Since you can't (correctly) talk about that infinite sum you like, the trick is to stick to finite cases and take the limit as the finite cases "approach" the infinite case.

For instance, it is true that:

<br /> \int_a^b f(x) \, dx = \lim_{n \rightarrow \infty}<br /> \sum_{i=0}^{n-1} (f(a + (i+1) \frac{b-a}{n}) - f(a + i \frac{b-a}{n})) \frac{b-a}{n}<br />

(fine print: this is only true if f is a bounded, Riemann-integrable function)

 

[ydnef]

Off course you are all more right than I am. I have just tried to be honest to myself, I did not want to deceive myself into believing that I really understood it all, like ''Look at me succesfully integrating sinus(x), so I must understand''. I tried to formulate into words my own thoughts on the subject as clear as possible, which is difficult. Sometimes you have to be more aggressive to learn something and keep asking why this and why that. Sometimes it is easier to fake understanding, which sometimes can be very easy, just repeat the language you've heard over and over again. Anyway, my mind is sort of a blank right now, do not know how to respond. I've learned something I guess, though it certainly did not satisfy my mind.

 

[Hurkyl]

It is difficult; it took a while before mathematicians came up with limits to allow them to rigorously deal with these things... and limits do seem awfully obtuse at first. But, the more you use them, the more sense they make.

And mathematicians like things to behave nicely too, so we define special classes of things that do behave nicely. For instance, an infinite series is "absolutely convergent" iff it's commutative and assocative. e.g.

1 + -1/2 + 1/4 + -1/8 + 1/16 + -1/32 + ...

is an absolutely convergent series becuase no matter how you rearrange and group these terms, you still get a sum of 2/3. However,

1 + -1 + 1 + -1 + 1 + -1 + ...

is not absolutely convergent, because it fails to be commutative and associative.

And it turns out that there is a simple criterion for a sequence to be in this class of functions (and this criterion is used as the definition): a series is absolutely convergent iff the series converges when you replace each term with its absolute value.


To relate this to what you said earlier, it turns out a (convergent) sequence is not absolutely convergent iff the sum of the positive terms and the sum of the negative terms are both divergent.

 

[ydnef]

Hey, this thread kind of died. Why? Can nobody think of anymore questions. Or is everything clear now to everybody. Maybe it just confused too much.

Anyway, here is one thought I would like to share. I first encountered quantum mechanics in a book where one would derive the Schrodinger equation using the notion of particles as wavepackets, and then afterwards the author would cleverly discard the derivation and keep the equation, arguing that the notion of a wavepacket may be useful to develop some intuition, but that in the end the idea cannot be taken too serious, and that everything in fact was far more general and abstract. Was the author doing first-time students of quantum mechanics an incredible favour or was he doing some unrepearable damage to them?

ydnef

 

[StonedPanda]

I've read the whole thread. I took AP Calc BC, so I have a year of calc under my belt (I got an A, so I guess that means I understood it). However, this is a question I'm still asking. WHY does the antiderivitive of a function give the area under the curve? I think this question is different from the one originally proposed -- i.e my question is different than why an infinite Reimann converges to an anti-derivitive. Or maybe I missed something... It's kinda late at night and I'm tired -- can anyone help? =p

 

[matt grime]

You have missed something - it is the fundamental thorem of calculus in action and has been explained in this thread, i think i did it twice.

 

[ydnef]

For developing some intuitive feel, I surely do recommend my own reflections on the matter, but be careful because this punk-kid of an ydnef has been abusing a lot of terminology and notation, which even confuses ydnef when he is not careful.

 

[matt grime]

That would be your system where you add up a countable number of (non-real) infinitesimals and get 1? Yep, that's the way forward...

 

[ydnef]

Yes you told me already that professional mathematicians have already reserved the word infinitesimal for some other mathematical idea and so forth. Got that. So okay, let's not call my dx's infinitesimals anymore, and think of every dx that appears as actually truly meaning ''limit dx->0''. Think of it part of me being too lazy and part of me wanting to keep the notation as tidy as possible.

I happen to think that it is useful in some practical cases to think of integration as some sort of ''adding of infinite terms'', like in some cases I like to think of matter as of being made out of point particles, though it does not take a genius to see that actually particles cannot really be points, they are more complex, and in some sense, nobody really really knows the true nature of matter, yet, but if you want to explain to a person for instance how television works, thinking of electrons as point particles is good enough. I know there is a flaw in that, but as long as it is not fatal, then it is not too bad I guess.

 

[matt grime]

You could of course just learn what the definition of integration is, and understand it properly, which might, just might, be considered the best way of doing it, seeing as it is the limit of finite sums approximating the area of the curve...

 

[ydnef]

You know what I've been thinking. That guy who invented/discovered non-standard analysis STOLE the term 'infinitesimal', not me. I mean, the term has been in existence ever since the time of leibniz and Newton, and who ever applied calculus between the time of its birth and somewhere between 1960 must have already had some sort of understanding connected to that word 'infinitesimal'. Hence, perhaps, all my confusion.

 

[mathwonk]

I am trying to discuss the answer to the question in thread 38, why is the FTC true? i.e. why does the antiderivative (of the height) give the area under the curve?

An equivalent question is why is the derivative of the area equal to the height of the curve?

The easiest way for me to understand anything is in a simple example. So take a constant function y = f(x) = c, for all x between a and b.

Then the area function A(x) = the area under the "curve" y = C, between a and x, is height times base = C times x-a = C(x-a) = Cx - Ca. So the derivative of this area function is C = height! So it is true in this case.


Now the next simplest case is a piecewise constant function, say y = C for x between a and r, and y = D for x between r and b, with a< r < b. let f(r) = D say (it does not matter).

Then the area function A(x) = C(x-a) for x between a and r, and equals

A(x) = C(r-a) + D(x-r), for x between r and b.

Thus the derivative of A exists except at r, and the derivative of A for x between a and r is C = the height, and the derivative of A for x between r and b is D = height.

And A is continuous. So here we are allowing as an "antiderivative" a function which is continuous everywhere, and differentiable where the original function is continuous, and has derivative equal to the height when the height is continuous. At those points where the original function is not continuous, we take the antiderivative to be whatever makes it continuous.

Note however that we have: total area = C(r-a) + D(b-r) = A(b)-A(a)
= C(r-a) + D(b-r) = the difference of the values of A at the endpoints a and b.


Now this continues to be true for all piecewise constant functions.

Moreover this property is preserved under uniform limits, so since every continuous function is a uniform limit of piecewise constant functions, and the antiderivatives also converge uniformly, it is still true for continuous functions.

I.e. if f is any continuolus function, and if A is an antiderivative of f, then the area under f equals A(b) - A(a). or equivalently, if A(x) is the area function from a to x, then A'(x) = f(x).

does this help? I realize it ain't full, but sometimes partial explanations help more than full ones.

 

[Ethereal]

I found the following interactive proof for the FTC:

http://archives.math.utk.edu/visual.calculus/4/ftc.9/

But for some reason, after looking through the whole proof, I still don't know why the anti-derivative of an integrable function for a certain range gives you the area under the graph. They said "Let A(x) = \int_a^x f(t) dt where A(x) is the area from x=a to x=t" But why is it possible for the area under the graph to be expressed as the anti-derivative in the 1st place?

 

[HallsofIvy]

You are missing the point.

A(x) = \int_a^b f(t) dt is defined as "the area bounded by y= f(x) above (assumed to be positive), y= 0 below, x= a on the left, and x= b on the right", not as an "anti-derivative".

The proof you give then uses the basic properties of area (if A and B are disjoint sets, then the area of A U B is area(A)+ Area(B)) to show that lim (A(x+h)-A(x))/h IS f(x) and so A itself IS an anti-derivative.

 

[mathwonk]

mr ethereal, have you read my post #45 above? I have tried to lay out the proof of FTC as simply as possible. I.e. it is true for piecewise constant functions, and hence also for their uniform limits, hence for all continuous functions. What do you think of that?

 

[Ethereal]

Actually, I don't follow your proof at all. I haven't studied maths formally yet, so I have difficulty understanding what a "piecewise constant function" is etc. Apparently a quick visit to mathworld.wolfram.com didn't help at all. Sorry.

EDIT: I think I just found out what those terms meant. I also happened to find this on the internet which seems similar to yours:

http://www.ma.hw.ac.uk/~robertw/F11UB3/slides1.pdf

 

[PaulScheduler]

Proofs For Volume

The method I know of is based on the principle of revolving a function around an axis. This is accomplished in one of two ways by using integration. The first is based on the area of a circle Vs the area of a cylinder. Often both can be used when set up appropriately.

I always pick the area of the circle method, whenever possible. This is because I see the function best that way.

Sphere:

Using the equation for a circle Y^2 + X^2 = R^2

Solve for Y


Replace the function f(x) or Y equal into the eqn pi * F(x)^2 , the area of a circle

This revolves any function around the X axis, summing up all areas, resullting in a volume

answer should be 4/3 pi R^3 Volume


2) Do the same for a line segment, except revolve it to get the volume of a cone

3) Volume of a cylinder is Y= a constant the simplest


Then take the definiete integral from 0 to R over dx