# Integration vs. Summation: What's the Difference?

Edited by Huma Saeed || By Sawaira Riaz || Updated on October 7, 2023
Integration involves combining parts into a whole, while summation is the process of adding quantities together.

## Key Differences

At their core, Integration and Summation represent foundational concepts in mathematics. Integration, stemming from calculus, relates to determining the area under curves, linking the infinitesimally small pieces into a cohesive whole. Summation, in contrast, is about adding together a series of numbers or quantities, representing the total.
Sawaira Riaz
Oct 03, 2023
Integration serves as a pivotal tool in understanding change and movement, calculating areas, and solving differential equations. Whether considering the distance traveled by an object based on its speed over time or analyzing fluid dynamics, integration offers depth. Summation, with its more straightforward role, allows mathematicians to calculate the total of sequences or series, often denoted using the Greek letter sigma (∑).
Sawaira Riaz
Oct 03, 2023
Delving into the realms of advanced mathematics, one finds integration can be definite or indefinite, with the former providing a numeric answer and the latter a function. Summation, too, can be finite or infinite, with finite summations yielding a set number and infinite ones potentially converging to a specific value.
Sawaira Riaz
Oct 03, 2023
Outside of mathematics, "integration" finds usage in contexts like business mergers or social inclusivity, emphasizing unity and cohesion. "Summation" may refer to a conclusive summary or a wrap-up of events, but its mathematical implication remains predominant.
Sara Rehman
Oct 03, 2023

## Comparison Chart

### Definition

Combining parts into a whole
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Oct 03, 2023

### Primary Use

Calculating area under curves
Calculating the total of sequences or series
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Oct 03, 2023

Sawaira Riaz
Oct 03, 2023

### Associated Fields

Calculus
Arithmetic, Algebra
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Oct 03, 2023

### Type Variants

Definite and Indefinite
Finite and Infinite
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Oct 03, 2023

## Integration and Summation Definitions

#### Integration

The inclusion of individuals or groups into a community
The school promoted the integration of all students regardless of background.
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Oct 03, 2023

#### Summation

The final result of a series of actions or events
The play's climax was the summation of all preceding scenes.
Aimie Carlson
Oct 03, 2023

#### Integration

Describing something as consistent or harmonious
The integration of the design elements made the room feel cohesive.
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Oct 03, 2023

#### Summation

A conclusive summary
At the meeting's end, he provided a quick summation.
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Oct 03, 2023

#### Integration

A calculus operation determining area under curves
Through integration, he calculated the displacement of the moving car.
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Oct 03, 2023

#### Summation

The process of adding things together
The summation of all expenses gave the total cost.
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Oct 03, 2023

#### Integration

The act of combining parts into a whole
The integration of various departments streamlined the company.
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Oct 03, 2023

#### Summation

A mathematical operation denoting the total of a sequence
Using summation, she added the series of numbers together.
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Oct 03, 2023

#### Integration

The process of unifying systems or processes
Integration of the software systems improved efficiency.
Harlon Moss
Oct 03, 2023

#### Summation

A comprehensive representation
His speech was a summation of the entire project.
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Oct 03, 2023

#### Integration

The act or process of integrating.
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Oct 21, 2022

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Oct 21, 2022

#### Integration

The state of becoming integrated.
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Oct 21, 2022

#### Summation

A sum or aggregate.
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Oct 21, 2022

#### Integration

The bringing of people of different racial or ethnic groups into unrestricted and equal association, as in society or an organization; desegregation.
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Oct 21, 2022

#### Summation

A concluding argument after the presentation of a legal case, especially an argument made to a judge or jury by an attorney for a party as to why that party should prevail.
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Oct 21, 2022

#### Integration

(Psychology) The organization of the psychological or social traits and tendencies of a personality into a harmonious whole.
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Oct 21, 2022

#### Summation

(Physiology) The process by which multiple or repeated stimuli can produce a response in a nerve, muscle, or other part that one stimulus alone cannot produce.
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Oct 21, 2022

#### Integration

(Mathematics) The process of computing an integral; the inverse of differentiation.
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Oct 21, 2022

#### Summation

Summarization; summary; summing up.
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Oct 21, 2022

#### Integration

(Electronics) The process of placing more than one integrated circuit on a single microchip.
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Oct 21, 2022

#### Summation

(mathematics) summing; summing up; adding (adding up) of a series of items.
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Oct 21, 2022

#### Integration

The act or process of making whole or entire.
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Oct 21, 2022

#### Summation

The act of summing, or forming a sum, or total amount; also, an aggregate.
Of this series no summation is possible to a finite intellect.
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Oct 21, 2022

#### Integration

The process of combining with compatible elements in order to incorporate them.
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Oct 21, 2022

#### Summation

A concluding summary (as in presenting a case before a law court)
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Oct 21, 2022

#### Integration

(society) The process of fitting into a community, notably applied to minorities.
Integration into the city
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Oct 21, 2022

#### Summation

(physiology) the process whereby multiple stimuli can produce a response (in a muscle or nerve or other part) that one stimulus alone does not produce
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Oct 21, 2022

#### Integration

(US) racial integration.
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Oct 21, 2022

#### Summation

The final aggregate;
The sum of all our troubles did not equal the misery they suffered
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Oct 21, 2022

#### Integration

(calculus) The operation of finding the integral of a function.
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Oct 21, 2022

#### Summation

The arithmetic operation of summing; calculating the sum of two or more numbers;
The summation of four and three gives seven
Four plus three equals seven
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Oct 21, 2022

#### Integration

(biology) In evolution, the process by which the manifold is compacted into the relatively simple and permanent; supposed to alternate with differentiation as an agent in species' development.
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Oct 21, 2022

#### Integration

The act or process of making whole or entire.
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Oct 21, 2022

#### Integration

The operation of finding the primitive function which has a given function for its differential coefficient. See Integral.
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Oct 21, 2022

#### Integration

In the theory of evolution: The process by which the manifold is compacted into the relatively simple and permanent. It is supposed to alternate with differentiation as an agent in development.
Sawaira Riaz
Oct 21, 2022

#### Integration

The action of incorporating a racial or religious group into a community
Sawaira Riaz
Oct 21, 2022

#### Integration

The act of combining into an integral whole;
A consolidation of two corporations
After their consolidation the two bills were passed unanimously
The defendants asked for a consolidation of the actions against them
Sawaira Riaz
Oct 21, 2022

#### Integration

An operation used in the calculus whereby the integral of a function is determined
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Oct 21, 2022

## FAQs

#### Are Integration and Summation interchangeable?

No, they represent different mathematical processes.
Sawaira Riaz
Oct 03, 2023

#### What symbol represents Integration?

It's represented by the symbol ∫.
Aimie Carlson
Oct 03, 2023

#### What's the outcome of an indefinite integration?

It provides a function, not a specific number.
Harlon Moss
Oct 03, 2023

#### Does Summation always give a number?

Often, but infinite summations might converge to a value or diverge.
Sawaira Riaz
Oct 03, 2023

#### Can Summation be used outside of math?

Yes, it can mean a conclusive summary or wrap-up.
Sawaira Riaz
Oct 03, 2023

#### Is Integration exclusive to mathematics?

No, it's used in contexts like business and social settings.
Sawaira Riaz
Oct 03, 2023

#### How is integration used in physics?

It's pivotal for concepts like displacement, work, and energy.
Janet White
Oct 03, 2023

#### What is the main goal of summation in arithmetic?

To find the total or aggregate of a series of numbers.
Sara Rehman
Oct 03, 2023

#### Can integration be reversed?

Yes, differentiation is the reverse process.
Sara Rehman
Oct 03, 2023

#### What symbol is used for Summation?

The Greek letter sigma (∑) denotes it.
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Oct 03, 2023

#### Is every sequence summable?

No, not all sequences have a convergent summation.
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Oct 03, 2023

#### Is summation limited to real numbers?

No, it can be applied to complex numbers, functions, and more.
Sawaira Riaz
Oct 03, 2023

#### Does Integration always involve calculus?

In math, yes. It's a fundamental operation in calculus.
Harlon Moss
Oct 03, 2023

#### Can summation be applied to infinite series?

Yes, provided the series is convergent.
Sawaira Riaz
Oct 03, 2023

#### How is integration crucial in engineering?

It aids in solving problems related to areas, volumes, and other quantities.
Janet White
Oct 03, 2023

#### What's the outcome of a definite integration?

It yields a specific numeric value.
Janet White
Oct 03, 2023

#### What does summation imply in a narrative?

It suggests a conclusive wrap-up or summary.
Sawaira Riaz
Oct 03, 2023

#### Are integrals always solvable?

No, some functions lack elementary antiderivatives.
Sawaira Riaz
Oct 03, 2023

#### How does integration relate to differentiation?

They are inverse operations in calculus.
Harlon Moss
Oct 03, 2023

#### What's the primary tool for summation in math?

Sigma notation, represented by ∑.
Aimie Carlson
Oct 03, 2023