Which definition of the integral is more "mathematically rigorus"?

**mfetch22** · August 16th 2010, 12:10 PM

Which one of these defintions is considered more "mathematically rigourous"? And which one is easier to manipulate? That is, which one is used more often to manipulate in a manner that proves certain theorems? Or, is there a different definition that is used? Also, what other definitions of the integral are there then these? (And please don't hesitate to point out any errors in the definitions below, because I'm no expert

)

[A] $\int_a^b \; [\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}] \; dx = f(b) - f(a)$

[B] $\int_a^b \; f(x) \; dx = \lim_{n \to \infty} \sum_{i=1}^n [ f(x_i) \cdot (\frac{b-a}{n})]$

Thanks in advance

**Defunkt** · August 16th 2010, 01:26 PM

The first one is not the definition of the integral - it is a consequence of the definition.

There are two equivalent definitions for the Riemann integral, one using Darboux sums and the other using Riemann sums:

Darboux integral - Wikipedia, the free encyclopedia
Riemann sum - Wikipedia, the free encyclopedia

Thread: Which definition of the integral is more "mathematically rigorus"?

LinkBack

Thread Tools

Display

Which definition of the integral is more "mathematically rigorus"?

Similar Math Help Forum Discussions

Question about a definition in "Topology from the differentiable viewpoint" (Milnor)

Ellipse: extract "minor axis" (b) when given "arc length" and "major axis" (a)

Definition clarification: "state-space", "support" and "sample space"

"The end of Pi?", this can't be mathematically valid, right?

Rigorous Definition of "Inequality" or "Positive" in the Reals

Search Tags