Why can't the indefinite integral be written with sigma notation?

[hddd123456789]

Hi, I've been wondering this since I started learning integration. I get that ∫ is basically an elongated S for "sum", because that is what it is basically doing. But then Ʃ does the same thing as well. If I'm understanding the difference, it is that Ʃ increments by finite measures, whereas ∫ increments by infinitesimal measures. But even if that is the difference, can't you 'simulate' infinitesimal incrementation with sigma notation? I gave it a shot below:

$∫f(x)dx=\sum^{∞}_{i=1}f(idx)dx+\sum^{∞}_{i=1}f(-idx)dx=\sum^{∞}_{i=1}f(idx)dx+f(-idx)dx=\sum^{∞}_{i=-∞}f(idx)dx$

Which is basically saying that the indefinite integral is equal to a sum. The way to get the sum is to divide the real number line from 0 to ∞ into ∞ parts. I get that ∞ isn't any particular number, but shouldn't we at least be able to say that each of these ∞ parts has a length of dx?

If that's the case, then you would take two sums, one from 0 to ∞, and the other from 0 to -∞ and increment by dx as you sum each area given by f(i*dx)*dx from the positive and negative ends of the number line. What do you think?

[Edit: I see that the first two sums is actually not equal to the third, since the first two actually skip over 0. So technically, the indefinitely integral is equal to the last sum, not the first two?]

 

[jbriggs444]

The serious problem with the formula you give is that the result of an indefinite integral is supposed to be a function. The formula you give evaluates to a number.

The less serious problem is the one you already pointed out: ∞ is not a number. That caveat aside, the approach you give is the essense of the Riemann integral.

http://en.wikipedia.org/wiki/Riemann_integral

 

[hddd123456789]

Yeah, as soon as I tried to solve out using that formula, I realized for myself that this is basically trying to sum up the whole are under the curve to infinity. Basically it is equal to the definite integral from -∞ to ∞, as below?

$∫^{∞}_{-∞}f(x)dx=\sum^{∞}_{i=-∞}f(idx)dx$

 

[hddd123456789]

And actually, I had another question that was related enough that I figured I would ask in the same thread. I sort of understand, I think, that the motivation for calculating the area under the curve and determining of the anti-derivative/indefinite integral were not necessarily related. And that the indefinite integral said something about the area under the curve wasn't known a priori?

Well, anyway, what I'm wondering is when I see for example the following:

$f(x)=\frac{d}{dx}F(x)=\frac{F(x+dx)-F(x)}{dx}$

I can apply it algebraically using infinitesimals to, for example, f(x)=x^2, as per:

$\frac{d}{dx}x^2=\frac{(x+dx)^2-x^2}{dx}$
$=\frac{x^2+2xdx+dx^2-x^2}{dx}$
$=\frac{2xdx+dx^2}{dx}$
$=\frac{dx(2x+dx)}{dx}$
$=2x+dx$
$=2x+0$
$=2x$

But is there an equivalent algebraic process for indefinite integration that applies at least to polynomials? I get that integration is basically like taking an infinitesimal sum, which from the scale of real numbers would be akin to taking an infinite sum of infinitesimals, so the only method that I can think of is the algebraic equivalence of a series. But these just seem so clunky for me. Am I missing something, or is the only practical way to determine the indefinite integral to just follow the derivation process in reverse?

And apologies if my terminology is a bit non-rigorous.

 

[CellCoree]

I can provide an explanation for why the indefinite integral cannot be written with sigma notation. The main reason is that sigma notation is used to represent a finite sum, while the indefinite integral represents an infinite sum.

In your example, you are attempting to divide the real number line from 0 to ∞ into an infinite number of parts, which is not possible. This is because the concept of infinity is not a number and cannot be divided into parts.

Furthermore, the notation used in integration, specifically the dx at the end, represents an infinitesimal change in the independent variable. This cannot be simulated with sigma notation because sigma notation is based on finite increments.

Moreover, the ∫ symbol represents the limit of a sum as the number of increments approaches infinity. This concept cannot be expressed with sigma notation, which only deals with finite sums.

In summary, while sigma notation and integration may seem similar in their use of summation, they are fundamentally different concepts and cannot be used interchangeably. The indefinite integral represents an infinite sum and cannot be written with sigma notation, which is used for finite sums.