definite integral

USAGE SUMMARY

The phrase "definite integral" is correct and usable in written English. You can use it in mathematical contexts, particularly in calculus when discussing integrals with specified limits. For example, "The definite integral of a function from a to b represents the area under the curve between those two points." Alternative expressions include "bounded integral" and "integral with limits."

✓ Grammatically correct

Academia

Science

Encyclopedias

Human-verified examples from authoritative sources

Exact Expressions

60 human-written examples

Topics: limits, rates of change, the derivative and applications, introduction to the definite integral and integration.

Stanford University

Inequality (14) is achieved by applying (15 - 18 15 - 18e definition of definite integrand(given in Secthen 2 above) to (19).

Journal of Inequalities and Applications

Influenced by the work of Émile Borel, Camille Jordan, and others, Lebesgue formulated a new theory of measure and framed a new definition of the definite integral, which he presented in his doctoral thesis at the Sorbonne in 1902.

Britannica

It is important to clarify the difference between the Monte Carlo integration of a definite integral as used here and the Monte Carlo simulation of a stochastic process [2].

EURASIP Journal on Wireless Communications and Networking

Methods of symbolic and numerical integration, applications of the definite integral, introduction to differential equations, infinite series.

Stanford University

Applications of differential calculus; introduction to integral calculus of functions of one variable, including: the definite integral, methods of symbolic and numerical integration, applications of the definite integral.

Stanford University

Because the definite integral of a function determines the area under its curve, integration is still sometimes referred to as quadrature.

Britannica

The concept of definite integral satisfies following properties: Let (F x)) and (G x)) be two integrable interval-valued functions.

Advances in Difference Equations

The Definite Integral.

Stanford University

Definite integral; fundamental theorem of calculus.

Massachusetts Institute of Technology

The definite integral of f from a to b.

Massachusetts Institute of Technology

Expert writing Tips

Best practice

When using "definite integral", always specify the limits of integration to ensure clarity and avoid ambiguity. For example, state "the definite integral from a to b" instead of just "the definite integral".

Common error

Avoid using "definite integral" when you're referring to the general antiderivative. The "indefinite integral" represents a family of functions, while the "definite integral" yields a specific numerical value representing area or accumulation.

Linguistic Context

The phrase "definite integral" functions as a noun phrase, typically acting as the subject or object in mathematical statements. It denotes a specific type of integral calculus operation, as illustrated by Ludwig's examples showcasing its use in defining areas and solving equations.

Expression frequency: Very common

Frequent in

Science

40%

Academia

35%

Encyclopedias

25%

Less common in

News & Media

0%

Formal & Business

0%

Wiki

0%

Ludwig's WRAP-UP

The phrase "definite integral" is a fundamental concept in calculus, widely used across science, academia, and encyclopedic references. Ludwig confirms its grammatical correctness and high frequency, especially in formal contexts. Understanding its distinction from the "indefinite integral" and specifying integration limits are key for precise communication. Related terms include "certain integral", "bounded integral", and "Riemann integral", each offering nuanced perspectives. By grasping these aspects, writers can confidently use "definite integral" in their mathematical and scientific discourse.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

Certain integral

This alternative is a more descriptive way to express the same mathematical concept.

Bounded integral

Focuses on the integral having defined limits, implying a finite area or value.

Integral with limits

Emphasizes the presence of upper and lower bounds for integration.

Riemann integral

Specifies the method of integration based on Riemann sums.

Lebesgue integral

Identifies the integral type based on Lebesgue's measure theory.

Numerical integration

Highlights the process of approximating the value of a definite integral.

Symbolic integration

Refers to finding a closed-form expression for the integral.

Quadrature

An older term for numerical integration, often used in specific contexts.

Area under a curve

A geometrical interpretation of the definite integral.

Fundamental theorem of calculus

Relates definite integrals to antiderivatives.

FAQs

How is the "definite integral" used in practice?

The "definite integral" is used to calculate the area under a curve between two points, and it has wide applications in physics, engineering, and economics. It can determine displacement, work, or total cost, depending on the function being integrated.

What's the difference between "definite integral" and "indefinite integral"?

A "definite integral" has upper and lower limits and results in a numerical value, representing the area under a curve between those limits. An "indefinite integral" does not have limits and results in a function that represents the antiderivative of the original function, plus a constant of integration.

What are some methods for evaluating a "definite integral"?

Common methods for evaluating a "definite integral" include using the Fundamental Theorem of Calculus, substitution, integration by parts, and numerical methods like Simpson's rule or the trapezoidal rule. The choice of method depends on the complexity of the integrand.

Can a "definite integral" be negative?

Yes, a "definite integral" can be negative. This occurs when the function being integrated is below the x-axis for a portion of the interval of integration, or when the upper limit of integration is less than the lower limit.