## Cardinalic flaw of Riemann integral

I have learnt 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.
 PhysOrg.com science news on PhysOrg.com >> City-life changes blackbird personalities, study shows>> Origins of 'The Hoff' crab revealed (w/ Video)>> Older males make better fathers: Mature male beetles work harder, care less about female infidelity
 Recognitions: Science Advisor 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.
 Recognitions: Science Advisor Each Δxi is a continuum - there is no contradiction.

## Cardinalic flaw of Riemann integral

 Quote by pyfgcr I have learnt 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.

Recognitions:
Gold Member
Staff Emeritus
 Quote by pyfgcr 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$^{n=1}_{∞}$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 suprise that your mistaken definition is not valid.

 Similar discussions for: Cardinalic flaw of Riemann integral Thread Forum Replies Calculus & Beyond Homework 0 Calculus & Beyond Homework 2 Calculus & Beyond Homework 1 Calculus & Beyond Homework 0 Introductory Physics Homework 1