iterated integral
C2 / Highly TechnicalFormal, technical, academic
Definition
Meaning
A multiple integral in mathematics where integration is performed with respect to one variable at a time in a specified order.
In broader applications, it can refer to the result of repeatedly applying the process of integration, or metaphorically to any deeply nested or layered process of cumulative calculation or refinement.
Linguistics
Semantic Notes
This is a specialized term from mathematical analysis. It implies a sequence of definite integrals, each taking the result of the previous integration as part of its integrand or limits. The order of integration is crucial.
Dialectal Variation
British vs American Usage
Differences
No significant differences in meaning or usage. Pronunciation may vary slightly (see IPA). Spelling of mathematical terms like 'formulae' (BrE) vs. 'formulas' (AmE) in surrounding text is a potential difference.
Connotations
Identical technical connotations in both varieties.
Frequency
Used with identical low frequency in university-level mathematics in both regions.
Vocabulary
Collocations
Grammar
Valency Patterns
iterated integral over [region R]iterated integral of [function f(x,y)]iterated integral with respect to [x then y]Vocabulary
Synonyms
Strong
Neutral
Weak
Vocabulary
Antonyms
Phrases
Idioms & Phrases
- “none”
Usage
Context Usage
Business
Almost never used.
Academic
Core term in university-level calculus, multivariable calculus, and mathematical physics courses.
Everyday
Virtually non-existent.
Technical
Used precisely in mathematics, physics (e.g., quantum mechanics, electromagnetism), and engineering disciplines involving continuum mechanics.
Examples
By Part of Speech
verb
British English
- We must iterate the integration process to solve for the volume.
- The solution involves integrating first with respect to x and then iterating that operation for y.
American English
- You need to iterate the integral to account for all three dimensions.
- The function is iteratedly integrated across the specified domain.
adverb
British English
- The function was integrated iteratively.
- The solution is found by integrating iteratively over each variable.
American English
- Apply the operator iteratively to build the solution.
- The calculations proceed iteratively from the innermost integral.
adjective
British English
- The iterated integral approach is standard here.
- We obtained an iterated integral expression for the mass.
American English
- The iterated integral form is more convenient for computation.
- His proof relied on an iterated integral representation.
Examples
By CEFR Level
- This word is too advanced for A2 level learners.
- This word is too advanced for B1 level learners.
- In our maths class, we started learning about integrals over areas, which are called iterated integrals.
- To find the volume, you calculate an iterated integral.
- Fubini's Theorem allows us to evaluate a double integral by expressing it as an iterated integral in either order, provided the function is continuous.
- The challenge was to correctly establish the limits for the iterated integral after switching the order of integration.
Learning
Memory Aids
Mnemonic
Imagine peeling an onion layer by layer. Each integration solves for one variable's 'layer', leaving the next one to be integrated, just as you remove one onion skin to reveal the next.
Conceptual Metaphor
LAYERED ACCUMULATION: A process of summing up quantities in a stepwise, hierarchical manner, where each step builds upon the condensed result of the previous one.
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Avoid direct translation that yields 'повторенный интеграл' as a primary term; the standard Russian equivalent is 'кратный интеграл' or more specifically, 'повторный интеграл'.
- Confusion with 'интеграл итерации' (integral of iteration) is possible but incorrect. The concept is of iteration *of* the integration operation.
Common Mistakes
- Incorrectly changing the order of integration without adjusting the limits, leading to a wrong answer.
- Using indefinite integrals at an inner step instead of definite integrals with correct variable limits.
- Pronouncing 'iterated' as 'eye-ter-ated' instead of 'it-er-ated'.
- Treating the variables as independent after the first integration, forgetting the inner result is a function of the outer variable.
Practice
Quiz
What is the defining characteristic of an iterated integral?
FAQ
Frequently Asked Questions
A double integral is a definite integral over a two-dimensional region. An iterated integral is one practical method to compute a double (or multiple) integral by performing successive one-dimensional integrations.
Yes, the order is crucial. While Fubini's Theorem states that under certain conditions (like continuity) the final result is the same regardless of order, the limits of integration and the difficulty of computation can change dramatically with the order.
Yes, absolutely. Iterated integrals extend to any finite number of dimensions (triple for volume, quadruple for hypervolume, etc.). The process is conceptually the same: integrate sequentially with respect to each variable.
No. While foundational in pure maths (analysis), it is a fundamental tool in applied fields like physics (e.g., calculating centre of mass, moment of inertia), engineering, probability theory (for joint probability densities), and economics in complex modelling.