The Fundamental Theorem of Calculus

[dx]

Id like to know if the following argument is valid.
Take an arbitrary function $$f(x)$$. $$f(x)dx$$ can be thought of an infinitesimal area of a certain form (I emphasise this because I use it later in the argument) determined by the form of the function $$f(x)$$. Let's denote its integral by $$Y$$.

$$\int{ f(x)} dx = Y$$

Now, I argue that an infinitesimal of the whole formed by summing up infinitesimals of a certain form must be an infinitesimal of the same form. ie,

$$d\int{ f(x)} dx = f(x)dx$$

Then the fundamental theorem of calculus follows.

$$f(x) dx = dY$$

$$f(x) = \frac{dY}{dx}$$

$$f(x) = \frac{d}{dx}\int{ f(x)} dx$$

If this argument is valid. Can it be made rigorous?

 

[matt grime]

No, it is not valid. If f is arbitary there is no reason for its integral to be differentiable, and indeed trivial examples prove this.

 

[dx]

But I am not differentiating it. I am just saying that I am taking an infinitesimal part of $$Y$$. Any way, let's say both f(x) and its integral are differentiable. Then is it valid?

EDIT : even if it is not completely valid, is it atleast an argument that suggests the fundamental theorem of calculus? I am talking about his particular statement.

"Now, I argue that an infinitesimal of the whole formed by summing up infinitesimals of a certain form must be an infinitesimal of the same form."

Is it ok to think this way, or is there something really really wrong with the way I understand it?

 

[matt grime]

But I am not differentiating it.

By it we are referring to the integral, and you are definitely differentiating that.


As for the rest, I struggle to decipher the words that have necessary meaning mathematically.

 

[dx]

Im trying to say that when you concatenate a 'd' on the left side of some quantity, you are making it an infinitesimal. So, we have an integral which is summing up infinitesimal quantities. So, in a way, the integral in made up of infinitesimals of a certain form ($$x^{2}dx$$,or $$e^{x}dx$$ etc.) When you concatenate a d to the integral, you are undoing the summing up and getting back the infinitesimal that you originally summed up. so $$d\int{f(x)}dx = f(x)dx$$.

 

[matt grime]

Im trying to say that when you concatenate a 'd' on the left side of some quantity, you are making it an infinitesimal.

you might be, I'm not sure the rest of mathematics would support that view in the way you wish it to.

So, we have an integral which is summing up infinitesimal quantities.

no, we don't, not really, you're conflating an analogy with the actual thing itself

So, in a way, the integral in made up of infinitesimals of a certain form

again, no it isn't.

($$x^{2}dx$$,or $$e^{x}dx$$ etc.) When you concatenate a d to the integral, you are undoing the summing up and getting back the infinitesimal that you originally summed up.

no you're not.

 

[mathphys]

Let's say that f(x) is Riemann integrable, ok? EDIT: In some interval (a,b)

You take the limit of a Riemann sum to obtain its integral, with all that this implies, not by summing up 'infinitesimals of certain form'.

 

[dx]

ok, thanks.