# Integration Basics: Axioms & Answers

• MHB
• roni1
In summary, the basic axioms of an integration are the axioms of a field. Beyond that we have the definition of a derivative, and the fundamental theorems of calculus that follow from those axioms and definitions.
roni1
One of my students ask me:
"Which axioms are the basic of the integration?"
Any ideas?

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.

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

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.

Why "we don't actually need division"?

roni said:
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).

## 1. What is integration in mathematics?

Integration is a mathematical concept that involves finding the area under a curve. It is the reverse process of differentiation and is used to find the total value or quantity of a function over a given interval.

## 2. What are the basic axioms of integration?

The basic axioms of integration are the additivity property, the linearity property, and the continuity property. The additivity property states that the integral of a sum of two functions is equal to the sum of their individual integrals. The linearity property states that the integral of a constant times a function is equal to the constant times the integral of the function. The continuity property states that a function must be continuous in order for it to be integrable.

## 3. What is the fundamental theorem of calculus?

The fundamental theorem of calculus is a theorem that connects the concepts of differentiation and integration. It states that if a function is continuous on a closed interval [a, b] and F(x) is its antiderivative, then the definite integral from a to b of the function f(x) is equal to F(b) - F(a).

## 4. What are some common integration techniques?

Some common integration techniques include integration by substitution, integration by parts, and partial fraction decomposition. Integration by substitution involves substituting a variable with a new one in order to simplify the integral. Integration by parts is used to integrate products of functions, while partial fraction decomposition is used to integrate rational functions by breaking them down into simpler fractions.

## 5. How can I check my answer when performing an integration?

You can check your answer by taking the derivative of the result. If the derivative is equal to the original function, then the integration was performed correctly. You can also use online integration calculators or graphing software to plot the function and the integral and see if they match up.

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