The types of integrals that we find in the calculation are: Integrals Indefinite and Integrals Defined. Although the definite integrals have many more applications than the indefinite integrals, it is necessary to first learn to solve indefinite integrals.
One of the most attractive applications of the defined integrals is the calculation of the volume of a solid of revolution.
Solid Revolution
Both types of integrals have the same properties of linearity and also the techniques of integration do not depend on the type of integral.
But in spite of being very similar, there is a main difference; in the first type of integral the result is a function (which is not specific) whereas in the second type the result is a number.
Two Basic Types of Integrals
The world of integrals is very broad but within this we can distinguish two basic types of integrals, which have a great applicability in everyday life.
1- Integrals Indefinite
If f '(x) = f (x) for all x in the domain of f, we say that F (x) is an antiderivative, a primitive, or an integral of f (x).
On the other hand, we observe that (F (x) + C) '= F' (x) = f (x), which implies that the integral of a function is not unique, because giving different values to the constant C we will obtain different antiderivatives.
For this reason F (x) + C is called the Indefinite Integral of f (x) and C is called the integration constant and we write it as follows
Integral Indefinida
As we can see, the indefinite integral of the function f (x) is a family of functions.
For example, if we want to calculate the indefinite integral of the function f (x) = 3x2, we must first find an antiderivative of f (x).
It is easy to note that F (x) = x³ is an antiderivative, since F '(x) = 3x². Therefore, it can be concluded that
∫f (x) dx = ∫3x²dx = x³ + C.
2 - Defined integrals
Let y = f (x) be a real function, continuous in a closed interval [a, b] and let F (x) be an antiderivative of f (x). It is called the definite integral of f (x) between the limits a and b to the number F (b) -F (a), and is denoted as follows
Fundamental Theorem of Calculus
The formula shown above is best known as"The Fundamental Theorem of Calculus". Here"a"is called the lower limit and"b"is called the upper limit. As you can see, the definite integral of a function is a number.
In this case, if the defined integral of f (x) = 3x² in the interval [0,3] is calculated, a number will be obtained.
To determine this number we choose F (x) = x³ as the antiderivative of f (x) = 3x². Then we calculate F (3) -F (0) which results in 27-0 = 27. In conclusion, the definite integral of f (x) in the interval [0,3] is 27.
It is possible to emphasize that if G (x) = x³ + 3 is chosen, then G (x) is an antiderivative of f (x) other than F (x), but this does not affect the result since G (3) 0) = (27 + 3) - (3) = 27. For this reason, in the defined integrals the integration constant does not appear.
One of the most useful applications of this kind of integral is that it allows the calculation of the area (volume) of a flat figure (of a solid of revolution), establishing adequate functions and limits of integration (and a rotation axis).
Within the defined integrals we can find several extensions of this one such as line integrals, surface integrals, improper integrals, multiple integrals, among others, all with very useful applications in the sciences and engineering.
References
- Casteleiro, J. M. (2012). Is it easy to integrate? Self-learning manual. Madrid: ESIC.
- Casteleiro, J. M., & Gómez-Álvarez, R. P. (2002). Integral calculus (Illustrated ed.). Madrid: ESIC Editorial.
- Fleming, W., & Varberg, D. E. (1989). Precalculus Mathematics. Prentice Hall PTR.
- Fleming, W., & Varberg, D. E. (1989). Precalculus mathematics: a problem-solving approach (2, Illustrated ed.). Michigan: Prentice Hall.
- Kishan, H. (2005). Integral Calculus. Atlantic Publishers & Distributors.
- Purcell, E.J., Varberg, D., & Rigdon, S. E. (2007). Calculation (Ninth ed.). Prentice Hall.