How are calculating infinite points?

And yet we get one definite answer that is the area. The logic makes sense except for how we are able to calculate the infinite f(x) values for the function

A Riemann sum is just a particular instance of an infinite series. You can prove that you can take an arbitrarily small minimum length of the rectangles and demonstrate that it is arbitrarily close to a certain sum and we take that to mean something. Because it does. It means something is true for every conceivable Riemann sum, so when you use something against any theoretical Riemann sum (which is implicitly done through standard applications of integration), the conclusion you made is true.

It isn't actually doing all that work, yes it would be impossible to do for for example
f(0.000000001)dx + f(0.00000000002)dx + ...

As a matter of fact the above would not work because the x values are infinitely small to get the precise area.

What the rienmann sum is saying is that this seemingly infinite work ends up actually being a limit as your n approaches infinity. that's what's happening.

>>9153698
>>9153670
I think the question was "how come increasing a finite approximation more and more gives all the information about this infinite object" and the answer is "because [a,b] is compact and the function is e.g. continuous"

aka mathemagic my man

>>9153710
Yeah but if those rectangles have no width they'd have no area. How can an infinite amount of zero width rectangles add up to any amount greater or lesser than 0?

>>9153670
>how are we calculating infinite points
by using limits, or least upper bounds

>>9153926
its because the limit of (x_n)*(y_n), where x_n approaches 0, and y_n approaches infinity, is an indeterminate form. An infinitely increasing set of shrinking rectangles could converge to area 0, or it could converge to a finite area, or it could converge to an infinite area. If you want to perform limit operations automatically, or if you want the analysis proofs to be simpler, you could look up hyperreal numbers, where the process of converging towards 0 can be quantified, and expressed with algebra.