The Decalogue to Set Up a Definite Integral

Consider the situation in which it is required to calculate a quality of an entire thing, a whole, but that quality depends on $x$ , and there is no direct formula to calculate it. This type of problems can be solved by means of the following procedure.

Step 1: To point out the whole of which the quality will be calculated.
Step 2: To express with the symbol $M$ the numerical value of the quality of that whole.
Step 3: To divide appropriately the whole into an infinitely large amount of parts
(each one infinitely small) so that in each part the quality required for the whole can be calculated.
Step 4: To point out one generic part that represents all the infinitely small parts in which the whole was divided.
Step 5: To express with the symbol $dM$ the numerical value of the quality of that generic part.
Step 6: To express $M$ as the sum of the values of $dM$ .
Step 7: To calculate the exact value of $dM$ .
Step 8: To express $dM$ in the form $r(x) \cdot dx$ .
Step 9: To express $M$ as a Definite Integral.
Step 10: To calculate the Definite Integral to obtain the numerical value of $M$ .

Recall that in order to apply this Decalogue first it is necessary to verify that the quality to be quantified may be expressed as depending on one variable. In the Decalogue, the variable used was $x$ , but it can be applied as well if the variable is $y$ (or any other, depending on the mathematical model) instead.

Let $y = F(x)$ be a continuous and differentiable function in the closed interval $[a,b]$ . In such interval, the quantity $dy = F'(x)\cdot dx$ is an infinitely small quantity by the definition of differential.

By the properties of the infinitesimals, the sum of any finite number of infinitesimals, is an infinitesimal. So that in order to obtain a finite number adding up differentials, it is neccesary to consider an infinite amount of terms in the sum. This process of adding up an infinite amount of infinitesimals is encapsulated in the concept of the Definite Integral.

Definite Integral

It is the result of the sum of an infinite amount of differentials and it is denoted by:

$\begin{equation*} \int\limits_{a}^{b} f(x) \cdot dx \end{equation*}$

where $a$ and $b$ are called «the upper and lower limits of integration», respectively and these values are the begining and ending points of the evaluation of the sum.

The notation:

$\begin{equation*} \int\limits_{a}^{b} f(x) \cdot dx \end{equation*}$

is read: the definite integral of the function $f$ from $x = a$ to $x = b$ .

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The Decalogue to Set Up a Definite Integral

You will understand the idea of Definite Integral as a sum of an infinite amount of differentials.

Definite Integral