One of the most important branches of mathematics includes calculus. Calculus is a method of systematically calculating problems, which deals with finding properties or values of functions by integrals and derivatives.
Key Takeaways
- Definite integrals calculate the signed area under a curve within a specific interval, providing a numerical value.
- Indefinite integrals determine the antiderivative of a function, expressing the result as a family of functions with an added constant.
- Both definite and indefinite integrals are important concepts in calculus, but they serve different purposes: definite integrals quantify areas, while indefinite integrals explore antiderivatives.
Definite vs. Indefinite Integrals
The difference between Definite and Indefinite Integral is that a definite integral is defined as an integral that has upper and lower limits and has a constant value as the solution; on the other hand, an indefinite integral is defined as an 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 with upper and lower limits. An indefinite integral is the representation of a family of functions without limits.
Comparison Table
| Parameter of Comparison | Definite Integrals | Indefinite Integrals |
|---|---|---|
| What it means | A definite integral has lower and upper limits and, on solving, gives a constant result. | An indefinite integral is an integral with no limits, and a mandatory arbitrary constant is added to the integral. |
| What it represents | The definite integral represents the number when its upper and lower limits are constant. | An indefinite integral represents a family of various functions with derivatives f. |
| Limits applied | The upper and lower limits applied in a definite integral are always constant. | In indefinite integral, there are no limits since it is a general representation. |
| Solution obtained | The 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 with a constant value added to it, which is represented by C. |
| Used for | A definite integral is widely used in physics and engineering. Some of the areas of use of a definite integral include calculating values of force, mass, work, areas between curves, volumes, act length of curves, surface areas, moments and center of mass, exponential growth, decay, etc. | Indefinite integrals are used in fields like business and sciences, including engineering, economics, etc. It is used where a general solution is required for a problem. |
What is a Definite Integral?
A definite integral represents a number that gives a constant result. A definite integral always has an upper limit and a lower limit.
The solution can either be positive or negative. The solution obtained from a definite integral always lies in a specific area.
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, the center of mass, exponential growth and decay, etc.
What is Indefinite Integral?
An indefinite integral is defined as the integral without limits. The indefinite integral represents a family of various functions having derivative f.
The solution obtained by solving the unknown function of an indefinite integral is a generalized solution; therefore, it also has variables. The area of the solution of an indefinite integral is not specified.
Indefinite integrals are used where a general solution to the problem is required. Indefinite integrals are used in business, sciences, engineering, economics, etc.
Main Differences Between Definite and Indefinite Integral
- A definite integral can be defined as an integral with limits; conversely, an indefinite integral can be defined as an integral without limits.
- A definite integral represents the number with constant upper and lower limits. In contrast, an indefinite integral represents the general solution for a family of functions having derivative f.
- https://www.tandfonline.com/doi/abs/10.1080/10652469.2014.1001385
- https://www.koreascience.or.kr/article/JAKO200931559904911.page
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