Cardinalic flaw of Riemann integral

• pyfgcr
The integral is the limit of finite Riemann sums, not infinite sums.In summary, the Riemann integral is not the sum of infinite rectangles, but rather the limit of finite Riemann sums over a partition of the interval. The interval is continuous, so the number of rectangles should be uncountable, not countable. This shows that the mistaken definition is not valid.

pyfgcr

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

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:
Each Δxi is a continuum - there is no contradiction.

pyfgcr said:
I have learned that integral is the Riemann sum of infinite rectangle, that:
Ʃ$^{n=1}_{∞}$f(xi)Δxi = ∫$^{b}_{a}$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?

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.

pyfgcr said:
I have learned 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$^{n=1}_{∞}$f(xi)Δxi = ∫$^{b}_{a}$f(x)dx
However, I think that (a,b) is the continuous interval, so the number of rectangle should be c instead of $\aleph$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 surprise that your mistaken definition is not valid.