# integral

Pronunciation: US say "IN-tuh-gruhl" /ˈɪntəɡɹəl/; UK say "IN-ti-gruhl" /ˈɪntɪɡr(ə)l/

## Adjective

1. Forming an essential part of a whole, so that the thing is incomplete without it.
   - Simple: Forming a necessary part of a whole, so it cannot be left out.
   - _Volunteers were integral to the success of the entire festival._
   - _Her research played an integral role in the company's new design._
4. Relating to or involving integration in calculus.
   - Simple: In math, having to do with the process of finding an integral.
   - _Integral calculus provides tools for computing areas and volumes._
   - _The physicist took an integral approach to model the accumulated charge._
5. Relating to or consisting of integers; whole-numbered rather than fractional.
   - Simple: In math, being a whole number, or about whole numbers.
   - _The equation has only integral solutions, such as 2 and 5._
   - _We restricted the variable to integral values._
8. In commutative algebra, said of a ring element relative to a subring A: being a root of some monic polynomial with coefficients in A.
   - Simple: In higher math, being an answer to a special kind of equation built from numbers in a smaller set.
   - _Every element of the smaller ring is integral over the base ring._
   - _An algebraic integer is integral over the integers._

## Noun

2. One of the two basic operations of calculus, which finds quantities such as area, volume, or displacement by summing infinitely many infinitesimal parts.
   - Simple: A math operation that adds up many tiny parts to find a total such as an area or distance.
   - _She used an integral to find the total distance the car traveled._
   - _The integral sums countless thin slices to yield the area beneath the graph._
3. The numerical result of integrating a function over a given range, such as an area or total: a value that may be finite or infinite.
   - Simple: The final number found by integration, such as an area or a total amount.
   - _The integral over that interval converges to a finite value._
   - _Because the area was unbounded, the integral diverged to infinity._
6. The result of integrating a function without fixed limits, producing another function; the function's antiderivative.
   - Simple: A function found by reversing the process of finding a rate of change.
   - _Every integral of this kind includes an arbitrary constant._
   - _She computed the integral to recover the original function from its derivative._
7. Any of several rigorous mathematical definitions of integration, such as the Riemann integral or the Lebesgue integral.
   - Simple: A formal math version of integration, such as the Riemann or Lebesgue kind.
   - _The Lebesgue integral extends the older Riemann integral to a much broader class of functions._
   - _Mathematicians choose which integral to use depending on the functions involved._

## Adjective

9. Whole and undamaged; complete in every part. *(obsolete)*
   - Simple: Complete and not broken or damaged.
   - _The seal was still integral when the tomb was opened centuries later._
   - _He was relieved to find the manuscript integral after the flood._

## Noun

10. In Newtonian calculus, the fluent corresponding to a given fluxion: the historical counterpart of the modern antiderivative. *(obsolete)*
   - Simple: In Newton's old system, the quantity found by reversing a given rate of flow.

## Origin

Borrowed from Middle French integral, which came from Medieval Latin integrālis, ultimately from Latin integer meaning "entire": the same source as the word integer.

Source: DictionaryAI, https://dictionaryai.com/define/integral (written from Wiktionary data, CC BY-SA 4.0: https://creativecommons.org/licenses/by-sa/4.0/).