triple integral
Low (Specialized Technical)Formal, Academic, Technical
Definition
Meaning
A mathematical operation involving three successive integrations over a three-dimensional region, used to compute quantities like volume, mass, or other properties.
In vector calculus, an integral of a function over a three-dimensional domain; the natural extension of single and double integrals to three variables.
Linguistics
Semantic Notes
Specifically denotes integration with respect to three variables (e.g., dx dy dz). Its primary physical interpretation is the calculation of volume under a surface in space.
Dialectal Variation
British vs American Usage
Differences
No significant lexical differences. The mathematical concept and term are identical. Potential minor differences in spoken rhythm or emphasis.
Connotations
Purely technical, formal, and academic in both dialects.
Frequency
Used exclusively in advanced mathematics, physics, and engineering contexts. Frequency is tied entirely to the prevalence of these subjects in curricula and professional work.
Vocabulary
Collocations
Grammar
Valency Patterns
triple integral of (a function) over (a region)triple integral ∫∫∫ f(x,y,z) dVVocabulary
Synonyms
Neutral
Weak
Vocabulary
Antonyms
Usage
Context Usage
Business
Virtually never used.
Academic
Core term in multivariable calculus, vector calculus, and mathematical physics courses.
Everyday
Never used.
Technical
Essential in engineering (e.g., fluid dynamics, electromagnetics), physics (e.g., quantum mechanics, thermodynamics), and advanced mathematics.
Examples
By Part of Speech
noun
British English
- The triple integral presented a significant challenge for the students.
- We shall evaluate the triple integral using an appropriate change of variables.
American English
- Setting up the triple integral is often the hardest part of the problem.
- The triple integral calculated the mass of the irregularly shaped object.
Examples
By CEFR Level
- A triple integral can be used to find the volume of a solid shape.
- The notation for a triple integral involves three integral signs.
- To compute the centre of mass, we must evaluate a triple integral of the density function over the entire body.
- The problem was simplified by converting the triple integral from Cartesian to spherical coordinates.
Learning
Memory Aids
Mnemonic
Think of it as a 'triple scoop' of integration: first integration scoops along x, second along y, third along z, to fill up a 3D volume.
Conceptual Metaphor
VOLUME IS ACCUMULATED SLICES (A volume is conceptualized as the sum (integral) of infinitely many infinitesimal slices (areas), which are themselves sums of lines).
Watch out
Common Pitfalls
Translation Traps (for Russian speakers)
- Direct translation "тройной интеграл" is perfectly correct and standard.
Common Mistakes
- Confusing the order of integration limits.
- Misapplying Fubini's theorem for non-continuous functions.
- Incorrectly translating coordinate systems (Cartesian to cylindrical/spherical).
Practice
Quiz
Which of the following is a typical use of a triple integral?
FAQ
Frequently Asked Questions
A double integral integrates over a two-dimensional area (e.g., to find area or surface mass), while a triple integral integrates over a three-dimensional volume (e.g., to find volume or the mass of a solid object).
Yes, often, but with careful adjustment of the integration limits to describe the same three-dimensional region. This is governed by Fubini's theorem under suitable conditions.
Cylindrical coordinates and spherical coordinates are frequently used when the region of integration has circular or spherical symmetry, respectively, as they can greatly simplify the limits and the integrand.
It is a compound noun consisting of two words: 'triple' (an adjective or determiner) modifying the noun 'integral'. It is always written as two separate words.