Cardinalic flaw of Riemann integral

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I have learnt that integral is the Riemann sum of infinite rectangle, that:
Ʃ[itex]^{n=1}_{∞}[/itex]f(xi)Δxi = ∫[itex]^{b}_{a}[/itex]f(x)dx
However, I think that (a,b) is the continuous interval, so the number of rectangle should be c instead of [itex]\aleph[/itex]0 (cardinality of natural number N).
So I wonder whether there are some problem that this definition is not valid anymore.
 

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  • #2
Bacle2
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How so? The oo you're using is the countable infinity. An uncountable sum will

necessarily diverge , unless only countably-many are non-zero. Still, good

question.

Edit: after reading SteveL's comment, I guess I should be more precise:

The limit in the sum you describe is a limit as you approach countable infinity;

so you are selecting one point x_i* in each subinterval , and , as N-->oo (countable

infinity) there is a bijection between the number of rectangles and the x_i* you choose.

Since the x_i* are indexed by countable infinity, so are the rectangles.
 
Last edited:
  • #3
mathman
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Each Δxi is a continuum - there is no contradiction.
 
  • #4
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I have learnt that integral is the Riemann sum of infinite rectangle, that:
Ʃ[itex]^{n=1}_{∞}[/itex]f(xi)Δxi = ∫[itex]^{b}_{a}[/itex]f(x)dx
I'm a little confused about this definition. Typically the Riemann integral is the limit of Riemann sums, each one of which is a finite sum over a partition of the interval. Each partition is a finite set of subintervals.

There is no infinite sum such as you've notated. Is this a definition you saw in class or in a book?
 
  • #5
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Thanks for explanation, I have understood.
And I mean it's the limit of finite sum, but I am a bit lazy so I remove the limit part for convenience.
 
  • #6
HallsofIvy
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I have learnt that integral is the Riemann sum of infinite rectangle,
No, it isn't. It is a limit of Riemann sums, each of which involves a finite sum. That is not "the Riemann sum of infinite rectangles" which is not defined.
that:
Ʃ[itex]^{n=1}_{∞}[/itex]f(xi)Δxi = ∫[itex]^{b}_{a}[/itex]f(x)dx
However, I think that (a,b) is the continuous interval, so the number of rectangle should be c instead of [itex]\aleph[/itex]0 (cardinality of natural number N).
So I wonder whether there are some problem that this definition is not valid anymore.
It should be no suprise that your mistaken definition is not valid.
 

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