riemann integral: meaning, definition, pronunciation and examples
Very LowTechnical/Academic
Quick answer
What does “riemann integral” mean?
A method of defining the integral of a function on an interval by partitioning the interval into subintervals, evaluating the function at sample points within each subinterval, and taking the limit of the sum of the products of function values and subinterval lengths as the partition becomes infinitely fine.
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Pronunciation
Definition
Meaning and Definition
A method of defining the integral of a function on an interval by partitioning the interval into subintervals, evaluating the function at sample points within each subinterval, and taking the limit of the sum of the products of function values and subinterval lengths as the partition becomes infinitely fine.
In mathematical analysis, the Riemann integral is the standard definition of the definite integral for functions on a closed interval. It provides a rigorous foundation for calculating areas under curves and is a precursor to more advanced integration concepts like the Lebesgue integral. It is named after the German mathematician Bernhard Riemann.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Spelling conventions follow standard British/American patterns for surrounding text (e.g., 'analyse' vs. 'analyze').
Connotations
Identical technical connotations in both varieties.
Frequency
Equally low and specialized in both academic communities.
Grammar
How to Use “riemann integral” in a Sentence
The Riemann integral of [function] on [interval] is [value].[Function] is Riemann integrable on [interval].One defines/computes/approximates the Riemann integral.Vocabulary
Collocations
Examples
Examples of “riemann integral” in a Sentence
adjective
British English
- The Riemann integral definition is foundational.
- We studied Riemann integrable functions.
American English
- The Riemann integral approach is standard.
- We examined Riemann integrable functions.
Usage
Meaning in Context
Business
Never used.
Academic
Exclusively used in university-level mathematics courses (real analysis, advanced calculus) and related research papers.
Everyday
Never used.
Technical
Used in rigorous mathematical discussions, textbooks, and proofs concerning integration theory.
Vocabulary
Synonyms of “riemann integral”
Neutral
Weak
Vocabulary
Antonyms of “riemann integral”
Watch out
Common Mistakes When Using “riemann integral”
- Incorrectly capitalizing 'integral' but not 'Riemann' (e.g., 'Riemann Integral').
- Using 'Riemann integral' to refer to indefinite integrals or antiderivatives.
- Pronouncing 'Riemann' as /raɪˈmæn/ instead of /ˈriːmən/.
FAQ
Frequently Asked Questions
It is named after the German mathematician Bernhard Riemann (1826–1866).
No. A function must be bounded and its set of discontinuities must have Lebesgue measure zero to be Riemann integrable on a closed interval.
The Lebesgue integral is a more general definition of integration that can handle a wider class of functions and is based on measuring the size of sets where the function takes certain values, rather than partitioning the domain.
It provides the foundational, rigorous definition for the definite integral taught in calculus, formalising the intuitive idea of area under a curve and linking it to the limit process.
A method of defining the integral of a function on an interval by partitioning the interval into subintervals, evaluating the function at sample points within each subinterval, and taking the limit of the sum of the products of function values and subinterval lengths as the partition becomes infinitely fine.
Riemann integral is usually technical/academic in register.
Riemann integral: in British English it is pronounced /ˈriːmən ˈɪntɪɡrəl/, and in American English it is pronounced /ˈriːmɑːn ˈɪntəɡrəl/. Tap the audio buttons above to hear it.
Learning
Memory Aids
Mnemonic
Think of 'Riemann' as 'Ree-man' who 're-mans' (re-measures) the area under a curve by adding up thinner and thinner rectangles.
Conceptual Metaphor
CALCULATING AREA IS SUMMING SLICES; A FUNCTION'S TOTAL OUTPUT IS THE ACCUMULATION OF INFINITESIMAL CONTRIBUTIONS.
Practice
Quiz
What is the primary context for the term 'Riemann integral'?