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

Edited by Aimie Carlson || By Janet White || 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.
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 (∑).
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.
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.

## Comparison Chart

### Definition

Combining parts into a whole

### Primary Use

Calculating area under curves
Calculating the total of sequences or series

### Associated Fields

Calculus
Arithmetic, Algebra

### Type Variants

Definite and Indefinite
Finite and Infinite

## 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.

#### Summation

The final result of a series of actions or events
The play's climax was the summation of all preceding scenes.

#### Integration

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

#### Summation

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

#### Integration

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

#### Summation

The process of adding things together
The summation of all expenses gave the total cost.

#### Integration

The act of combining parts into a whole
The integration of various departments streamlined the company.

#### Summation

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

#### Integration

The process of unifying systems or processes
Integration of the software systems improved efficiency.

#### Summation

A comprehensive representation
His speech was a summation of the entire project.

#### Integration

The act or process of integrating.

#### Integration

The state of becoming integrated.

#### Summation

A sum or aggregate.

#### Integration

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

#### 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.

#### Integration

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

#### 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.

#### Integration

(Mathematics) The process of computing an integral; the inverse of differentiation.

#### Summation

Summarization; summary; summing up.

#### Integration

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

#### Summation

(mathematics) summing; summing up; adding (adding up) of a series of items.

#### Integration

The act or process of making whole or entire.

#### 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.

#### Integration

The process of combining with compatible elements in order to incorporate them.

#### Summation

A concluding summary (as in presenting a case before a law court)

#### Integration

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

#### 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

#### Integration

(US) racial integration.

#### Summation

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

#### Integration

(calculus) The operation of finding the integral of a function.

#### 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

#### 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.

#### Integration

The act or process of making whole or entire.

#### Integration

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

#### 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.

#### Integration

The action of incorporating a racial or religious group into a community

#### 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

#### Integration

An operation used in the calculus whereby the integral of a function is determined

## FAQs

#### Are Integration and Summation interchangeable?

No, they represent different mathematical processes.

#### What symbol represents Integration?

It's represented by the symbol ∫.

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

It provides a function, not a specific number.

#### Does Summation always give a number?

Often, but infinite summations might converge to a value or diverge.

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

Yes, it can mean a conclusive summary or wrap-up.

#### Is Integration exclusive to mathematics?

No, it's used in contexts like business and social settings.

#### How is integration used in physics?

It's pivotal for concepts like displacement, work, and energy.

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

To find the total or aggregate of a series of numbers.

#### Can integration be reversed?

Yes, differentiation is the reverse process.

#### What symbol is used for Summation?

The Greek letter sigma (∑) denotes it.

#### Is every sequence summable?

No, not all sequences have a convergent summation.

#### Is summation limited to real numbers?

No, it can be applied to complex numbers, functions, and more.

#### Does Integration always involve calculus?

In math, yes. It's a fundamental operation in calculus.

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

Yes, provided the series is convergent.

#### How is integration crucial in engineering?

It aids in solving problems related to areas, volumes, and other quantities.

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

It yields a specific numeric value.

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

It suggests a conclusive wrap-up or summary.

#### Are integrals always solvable?

No, some functions lack elementary antiderivatives.

#### How does integration relate to differentiation?

They are inverse operations in calculus.

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

Sigma notation, represented by ∑.