Diferença entre integrais definidos e indefinidos (com tabela)

One of the most important branches of mathematics includes cálculo. Calculus is a manner of calculation of problems in a systematic way which usually deals with finding properties or values of functions by integrals and derivatives. The basic concept of calculus is differentiation and integration. The two concepts can be defined as the inverse of one another. The inverse of integral is diferencial and the inverse of the differential is integral. On the basis of results given by integrals, they are categorized as definite and indefinite.

Definite vs Indefinite Integrals

The difference between Definite and Indefinite Integral is that a definite integral is defined as the integral which has upper and lower limits and has a constant value as the solution, on the other hand, an indefinite integral is defined as the internal which do not have limits applied to it and it gives a general solution for a problem.

A definite integral of a function of an unknown variable is the representation of a number which has upper and lower limits. An indefinite integral is the representation of a family of functions without limits.

Comparison Table Between Definite and Indefinite Integrals

Parâmetro de ComparaçãoDefinite IntegralsIndefinite Integrals
What it meansA definite integral is the one that has lower and upper limits and on solving gives a constant result.An indefinite integral is the integral in which no limits are applied and has a mandatory arbitrary constant added to the integral.
What it representsThe definite integral represents the number when its upper and lower limits are constant.An indefinite integral is a general representation of a family of various functions with derivatives f.
Limits appliedThe upper and the lower limits applied in a definite integral are always constant.In indefinite integral, there are no limits since it is a general representation.
Solution obtainedThe values or solutions obtained from definite integrals are constant, however, they can either be positive or negative.The solution of an indefinite integral is a general solution and it has a constant value added to it which is generally represented by C.
Usado paraA definite integral is widely used in physics and engineering. Some of the areas of use of a definite integral include calculation of values of force, mass, work, areas between curves, volumes, act length of curves, surface areas, moments and center of mass, exponential growth and decay, etc.Indefinite integrals are of use in fields like business, sciences including engineering, economics, etc. It is used in areas where a general solution is required for a problem.

What is a Definite Integral?

A definite integral is defined as the representation of a number which gives a constant result. A definite integral always has an upper limit and a lower limit. The limits of the definite integrals are constant. Sometimes it is said that a definite integral is an indefinite integral evaluated over lower and upper bound.

The value or solution obtained on solving the integrals by applying limits are constant, therefore, these integrals are named as definite. The solution can either be positive or negative. The solution obtained from a definite integral always lies in a specific area. 

A definite integral is used when a function has two limits over which it is evaluated. A definite integral is widely used in all areas of physics and engineering. Some areas where definite integrals are used are a calculation of work, force, mass, areas, surface areas, the area between curves, length of arcs, moments, center of mass, crescimento exponencial and decay, etc.

What is Indefinite Integral?

An indefinite integral is defined as the integral without limits. The indefinite integral is the representation of a family of various functions having derivative f.  There are no limits applicable in an indefinite integral.

The solution obtained on solving the unknown function of an indefinite integral is a generalized solution and therefore it also has variables in it. The area of the solution of an indefinite integral is not specified. 

Indefinite integrals are used at places where a general solution to the problem is required. Indefinite integrals are used in business, sciences, engineering, economics, etc. Some of the areas of application of indefinite integral includes deslocamento from velocity, velocity from aceleração, voltage across capacitoretc.

Main Differences Between Definite and Indefinite Integral

  1. A definite integral can be defined as the integral which has limits, on the other hand, an indefinite integral can be defined as the integral without limits.
  2. A definite integral is the representation of the number when it has upper and lower limits constant whereas an indefinite integral is the representation of the general solution for a family of functions having derivative f.
  3. In the case of definite integrals, there are upper and lower limits, on the other hand, indefinite integrals do not have any limits.
  4. The solutions obtained by applying limits in the case of definite integrals are constants, on the other hand, the solution of an indefinite integral is a general solution.
  5. The solution of a definite integral can either be positive or negative whereas it is not known in indefinite integrals.
  6. The general solution obtained from indefinite integrals has an arbitrary constant and it is compulsory to add a constant to it.
  7. Both the integrals are of utmost importance in engineering and sciences, however, definite integrals are used in cases where a constant value is required as an answer while indefinite integral is used in cases where a general solution is required.

Conclusão

The two types of integrals have their properties and functions which play an important role in solving problems. If a definite integral is solved firstly by using indefinite integrals and applying limits after that, then it may have some discontinuities.

Referências

  1. https://www.tandfonline.com/doi/abs/10.1080/10652469.2014.1001385
  2. https://www.koreascience.or.kr/article/JAKO200931559904911.page