Definition

1. Noun:
   • A specific numerical value representing the accumulation of a function's values over a defined interval: In calculus, a definite integral is the result of integrating a function between two specific limits, representing the net area under the curve of the function between those points.

Usage

• The definite integral is used to calculate quantities such as area, volume, displacement, and total accumulated value when given a rate of change.
• It is written with the integral symbol, the function, and the lower and upper limits of integration. For example: ∫ from to of .
• The result of a definite integral is a number, not a function plus a constant.

Examples

• Noun:
  • The definite integral of the velocity function gives the total displacement of the object.
  • To find the area under the curve from x=1 to x=4, you must evaluate the definite integral.
  • The Fundamental Theorem of Calculus provides a method for calculating a definite integral using antiderivatives.

Advanced Usage

• "Evaluating a definite integral": The process of computing the numerical value of the integral.
  • After setting up the integral, the next step is evaluating the definite integral.
• "Limits of integration": Refers to the lower and upper bounds ( and ) in the notation ∫ₐᵇ f(x) dx.
  • The limits of integration for this problem are 0 and π.

Variants and Related Words

• Indefinite Integral (n): The general antiderivative of a function, expressed as a family of functions plus an arbitrary constant (e.g., ∫ f(x) dx).
• Integral Calculus (n): The branch of calculus concerned with integrals and their properties.
• Integrand (n): The function being integrated (e.g., in ∫ f(x) dx).

Synonyms

• Numerical integral (in a specific computational context).
• Riemann integral (a specific method for defining the definite integral).

Related Phrases

• "Take the definite integral of": To perform the operation of definite integration on a function over an interval.
  • We need to take the definite integral of the growth rate to find the total population increase.
• "Compute the definite integral": Synonymous with evaluating the definite integral.
  • The software can compute the definite integral numerically.

Related Concepts

• Fundamental Theorem of Calculus: The theorem that links differentiation and integration, stating that the definite integral of a function can be found using its antiderivative.
• Area under a curve: A common geometric interpretation of the definite integral.
• Net signed area: Emphasizes that the definite integral accounts for areas below the x-axis as negative.

Noun

1. the integral of a function over a definite interval