Integration Basics: Axioms & Answers

[roni1]

One of my students ask me:
"Which axioms are the basic of the integration?"
What I should answer him?
Any ideas?

 

[S.G. Janssens]

What is the most appropriate answer probably depends on the course you are teaching.

What course is it that you are teaching?

ADDITION (no pun intended): Probably all reasonable notions of integral are based on 1. an "obvious" way to integrate a certain class of simple functions - which then becomes a definition and 2. a limiting procedure of sorts to extend the integral to a much wider class of functions, preferably in such a manner that certain desirable properties hold.

How this is done for a particular integral type can vary a lot: Compare e.g. the Riemann and Lebesgue integral, already for the "basic" case of real-valued functions. When domain and range of the functions involved are allowed to be more general, the number of possibilities increases. (For example, when the co-domain is a function space that admits different topologies.)

So, how to best answer your student would depend on his background knowledge.

 

[MarkFL]

I've moved this thread to our "Calculus" forum as I assume we are discussing integral calculus. :)

 

[I like Serena]

The axioms are the axioms of a field, giving us addition, multiplication, subtraction, and division.
Everything else are definitions and theorems.
Notably the definitions of limits and integrals (for which we don't actually need division).
Beyond that we have the definition of a derivative, and the fundamental theorems of calculus that follow from those axioms and definitions.

 

[roni1]

Why "we don't actually need division"?

 

[I like Serena]

Why "we don't actually need division"?

For a Riemann-Integral (usually intended when we refer to integral) we need to be able to evaluate the sum of rectangular areas:
$$\sum_{i=0}^{n-1} f(t_i) \left(x_{i+1}-x_i\right)$$
and take it to the limit.
As you can see this requires subtraction, multiplication, and addition, but not division.
That means that if we would have the axioms of a Ring instead of a Field, that the integral would still be well-defined.
(A Field has all axioms that a Ring has and more.)

Btw, I've just realized that we actually need an Ordered Field (or Ordered Ring), since we also need the comparison operator $<$ to define a partition $\{x_i\}$ with $x_i < x_{i+1}$, and we need it for the definition of a limit as well.

Either way, to define the derivative, we need to be able to evaluate:
$$\frac{f(x)-f(a)}{x-a}$$
Therefore we need division.

Note that at this point I'm leaving out the more exotic definitions of integrals (such as the Lesbesgue-Integral) and derivatives (such as the Formal Derivative).