double integral: meaning, definition, pronunciation and examples
Very LowFormal, Technical, Academic
Quick answer
What does “double integral” mean?
A type of integral where the integration is performed over a two-dimensional area in a plane, resulting in the volume under a surface.
Audio
Pronunciation
Definition
Meaning and Definition
A type of integral where the integration is performed over a two-dimensional area in a plane, resulting in the volume under a surface.
In mathematics, an integral of a function of two variables, typically denoted as ∬ f(x,y) dx dy, computed over a region R. It represents the signed volume between the graph of the function and the xy-plane over the specified region.
Dialectal Variation
British vs American Usage
Differences
No meaningful lexical or usage differences. The concept and notation are identical in both varieties of English. Pronunciation differences follow general AmE/BrE patterns for the component words.
Connotations
None beyond its technical meaning.
Frequency
Frequency is identically low and context-dependent, appearing exclusively in advanced STEM education and professional discourse.
Grammar
How to Use “double integral” in a Sentence
to evaluate a double integral [over a region R]to compute the double integral [of f(x,y)]the double integral [representing the volume]Vocabulary
Collocations
Examples
Examples of “double integral” in a Sentence
verb
British English
- We need to double-integrate the function over the specified region.
- You can double integrate by first treating y as a constant.
American English
- We need to double-integrate the function over the specified region.
- You can double integrate by first treating y as a constant.
adverb
British English
- The function was integrated double over the square.
- None
American English
- The function was integrated double over the square.
- None
adjective
British English
- The double-integral formulation is more elegant for this problem.
- We applied a double-integral approach.
American English
- The double-integral formulation is more elegant for this problem.
- We applied a double-integral approach.
Usage
Meaning in Context
Business
Not used.
Academic
Core concept in multivariable calculus, vector calculus, and applied mathematics courses at university level.
Everyday
Not used.
Technical
Essential in engineering (e.g., calculating centroids, moments of inertia), physics (e.g., electromagnetism, quantum mechanics), and probability theory.
Vocabulary
Synonyms of “double integral”
Strong
Neutral
Weak
Vocabulary
Antonyms of “double integral”
Watch out
Common Mistakes When Using “double integral”
- Confusing the order of integration limits, leading to incorrect evaluation.
- Forgetting that the differentials (dx dy) imply an order of integration unless the region is specifically addressed.
- Using single-integral techniques incorrectly for a double integral.
FAQ
Frequently Asked Questions
A single integral finds the area under a curve (a 1D accumulation), while a double integral finds the volume under a surface (a 2D accumulation over an area).
The limits describe the region R in the xy-plane over which you integrate. You must express the bounds for one variable (e.g., y) in terms of the other (e.g., x), or use constant limits for simple rectangular regions.
Often yes, thanks to Fubini's theorem, provided the function is continuous over the region. However, the ease of calculation can depend heavily on the chosen order of integration (dx dy vs dy dx).
Not exactly. A double integral is performed over a flat, planar region. A surface integral is a more general concept that can be performed over a curved surface in three-dimensional space, though it often involves evaluating a double integral parametrically.
A type of integral where the integration is performed over a two-dimensional area in a plane, resulting in the volume under a surface.
Double integral is usually formal, technical, academic in register.
Double integral: in British English it is pronounced /ˌdʌb.əl ˈɪn.tɪ.ɡrəl/, and in American English it is pronounced /ˌdʌb.əl ˈɪn.t̬ə.ɡrəl/. Tap the audio buttons above to hear it.
Phrases
Idioms & Phrases
- “None”
Learning
Memory Aids
Mnemonic
Think of a 'Double' layer: first integrate in one direction (e.g., dx), then integrate that result in the perpendicular direction (e.g., dy), stacking up slices to find the total volume.
Conceptual Metaphor
SUMMATION IS ACCUMULATION (extended to two dimensions). The integral sums infinitesimal contributions (areas dx dy) weighted by the function's height to accumulate a total volume.
Practice
Quiz
What does a double integral typically compute?