What is the Difference Between Definite and Indefinite Integrals?
🆚 Go to Comparative Table 🆚The main difference between definite and indefinite integrals lies in the context of the limits of integration.
Indefinite Integrals:
- An indefinite integral is a function that, when differentiated, gives the original function (f(x)).
- It has no specified limits of integration, and the constant of integration is arbitrary.
- The process of finding an indefinite integral is also called integration or integrating.
Definite Integrals:
- A definite integral has limits of integration, which are specified as part of the problem.
- The limit values define a specific area under the curve of the function (f(x)).
- To compute a definite integral, one must first find the antiderivative (indefinite integral) of the function and then evaluate it at the endpoints (a and b).
In summary, an indefinite integral represents a function that, when differentiated, gives the original function, while a definite integral represents a specific area under the curve of a function.
Comparative Table: Definite vs Indefinite Integrals
Definite and indefinite integrals are both types of integrals used in calculus, but they serve different purposes and are applied in different contexts. Here is a table highlighting the differences between them:
Feature | Definite Integral | Indefinite Integral |
---|---|---|
Definition | Calculates the area under a curve between two specific points (a and b) | Finds the general form of a function that, when differentiated, produces a given function |
Interval | Requires specifying the lower and upper limits of integration (a and b) | Does not involve specific intervals but aims to determine a family of functions that differ only by a constant |
Definite vs. Indefinite Integrals: Differences and Examples | Unlike definite integration, indefinite integration does not involve specific intervals but aims to determine a family of functions that differ only by a constant | The process of calculating the exact two given values (definites) |
Symbol | $$\int_a^b f(x) dx$$ | $$\int f(x) dx$$ |
In summary, definite integration calculates the area under a curve between two specific points (a and b), while indefinite integration finds the general form of a function that, when differentiated, produces a given function. Definite integration requires specifying the lower and upper limits of integration, while indefinite integration does not involve specific intervals but aims to determine a family of functions that differ only by a constant. Both definite and indefinite integration can be represented by the symbol $$\int$$, followed by the function to be integrated and the differential variable.
- Derivative vs Integral
- Definite Loop vs Indefinite Loop
- Definite vs Indefinite Articles
- Integration vs Differentiation
- Infinity vs Undefined
- Riemann Integral vs Lebesgue Integral
- Differentiation vs Derivative
- Derivative vs Differential
- Discrete Function vs Continuous Function
- Integration vs Summation
- Inclusion vs Integration
- Difference Equation vs Differential Equation
- Inductive vs Deductive
- Algebra vs Calculus
- Instinct vs Intuition
- Dedifferentiation vs Redifferentiation
- Elastic vs Inelastic
- Variable vs Constant
- Discrete vs Continuous Variables