definite integral
Low (Specialist/Technical)Formal, Academic, Technical (Mathematics, Physics, Engineering)
Definition
Meaning
The evaluation of the integral of a function over a specified interval [a, b], resulting in a numerical value representing the net area between the curve and the x-axis.
In calculus, it represents the limit of a Riemann sum, providing the exact signed area under a curve. In applied contexts, it can represent total accumulation, such as total distance traveled given a velocity function.
Linguistics
Semantic Notes
Contrasts with 'indefinite integral' (antiderivative). The result is a number, not a function. Implies fixed, known limits of integration.
Dialectal Variation
British vs American Usage
Differences
No significant lexical differences. Potential minor spelling preferences ('centre' vs. 'center') in surrounding text but not in the term itself.
Connotations
Identical technical meaning and prestige.
Frequency
Equally low frequency, confined to STEM education and professional contexts.
Vocabulary
Collocations
Grammar
Valency Patterns
The definite integral of [FUNCTION] from [LOWER LIMIT] to [UPPER LIMIT] is [VALUE].We need to evaluate the definite integral.This theorem relates the definite integral to the antiderivative.Vocabulary
Synonyms
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Virtually never used.
Academic
Core concept in calculus courses, physics (work, probability), and engineering.
Everyday
Extremely rare, only in highly specific educational discussions.
Technical
Fundamental in mathematical analysis, computational science, and quantitative fields.
Examples
By Part of Speech
verb
British English
- We need to integrate the function from zero to pi.
- After you've integrated, substitute the limits.
American English
- We need to integrate the function from zero to pi.
- After integrating, plug in the bounds.
adverb
British English
- The function is integrally related to its derivative.
- The area was calculated integrally over the region.
American English
- The function is integrally related to its derivative.
- The area was calculated integrally over the region.
adjective
British English
- The integral value was computed numerically.
- This is a standard integral form.
American English
- The integral value was computed numerically.
- This is a standard integral form.
Examples
By CEFR Level
- The definite integral gives the area under a curve.
- We wrote the definite integral with numbers at the top and bottom of the integral sign.
- To find the total distance, we evaluated the definite integral of the speed function.
- The definite integral from 1 to 4 of 2x dx equals 15.
- The physicist used a definite integral to compute the total charge distributed along the wire.
- The value of the definite integral was shown to be independent of the path taken, illustrating a conservative vector field.
Learning
Memory Aids
Mnemonic
DEFinite integral has DEFinite (specific) start and end points, giving a DEfinite number.
Conceptual Metaphor
The definite integral is a CONTINUOUS SUMMATION MACHINE or a NET AREA ACCUMULATOR.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- False friend with 'интеграл' which can be both definite and indefinite. Must specify 'определенный интеграл'.
- The concept of 'пределы интегрирования' (limits of integration) is crucial.
Common Mistakes
- Confusing it with the indefinite integral (forgetting the limits).
- Misplacing the limits of integration (upper vs. lower).
- Forgetting that the result is a constant, not a function.
- Incorrectly applying the Fundamental Theorem of Calculus.
Practice
Quiz
What does a definite integral calculate?
FAQ
Frequently Asked Questions
A definite integral is a number (or constant), representing the net area/accumulation.
A definite integral has limits and yields a number. An indefinite integral (antiderivative) has no limits and yields a family of functions (+C).
Yes. It represents *signed* area. If the curve is below the x-axis over the interval, the definite integral is negative.
The numbers 'a' (lower limit) and 'b' (upper limit) at the bottom and top of the integral sign, defining the interval over which the integration is performed.