Intaai wowonod montok do Ponompuuvan

'''Ponompuuvan''' (Boros Inggilis: ''Integration'') nopo nga' karalano mongintob do sompuuvan (''integral''). Ponompuuvan id gana' [[Sidsimban]] nopo nga' binalik do [[Pongungaban]], ii karalano mongintob do tungab (''derivative''). It sompuuvan koubasan pinoulud id duo kawo: '''sompuuvan au ninggatang''' (''indefinite integral''), om '''sompuuvan ninggatang''' (''definite integral''). Kinootusan do ponompuuvan au ninggatang nopo nga' iso govit (''expression'') di au kigatang, nga' pinogovit id ngaan do piipiro osimban (''variable''), om it kinootusan nopo do ponompuuvan ninggatang nga' gatang toi tintob do togom (''constant quantity'') kopio.

Id timpohon, ponompuuvan orotian sabaagi do pomoruhangan do linaab id saau do isoiso pampos (''function''). Om ii pongomoi tagayo do ponompuuvan, ii rinuhang Riemann (''Riemann sum'') nopo pointudan id waya mongutok diti.

== Wayagait ==
Ingkaa no i wayagait do sompuuvan ninggatang i kivohit mantad <math>x=a</math> gisom <math>x=b</math>:

<math>\int_{a}^{b}{f(x) \mathop{dx}}</math>.

Om montok sompuuvan au ninggatang, tu' mantad pongomoi dau aiso gia vohit di sompuuvan au ninggatang. Ingkaa no wayagait:

<math>\int{f(x) \mathop{dx}}</math>.

== Poomitanan ==
Tu' ponompuuvan nopo nga' binalik di pongungaban, milo tokou mamakai tuku pongungaban poinsunsuri. Montok pampos <math>f(x) = 3x^2 + 2,</math> okito dati' i pampos diti nopo koiyo do tungab. Mantad dii, tu' ponompuuvan mamalik pongungaban, nung sompuuvanon tokou <math>f(x)</math> pakayon tokou i tuku sundu nga' poinsambalik nogi, ingkaa<ref> https://openstax.org/books/calculus-volume-1/pages/a-table-of-integrals#:~:text=1.,%E2%89%A0%E2%88%921</ref>:
<math>\int{f(x) \mathop{dx}} = \int{3x^2 + 2 \mathop{dx}},</math>

<math> = \frac{3x^{2+1}}{2+1} + 2x + T = x^{3} + 2x + T</math>

om <math>T</math> nopo nga' i togom ponompuuvan (''constant of integration'').

== Pongomoi ==
Koubasan it ruhang Riemann ponokoimpohon do pongomoi ponompuuvan, kopongotus do sompuuvan Riemann (''Riemann integral''). Nga' haro piipiro pampos do au osompuuvan don Riemann (''non Riemann-integrable''). Iso pongomoi do sompuuvan suai nopo nga' it sompuuvan Lebesgue.

=== Pongomoi sompuuvan Riemann ===
It sompuuvan Riemann popoindalad do ruhang Riemann, maya' dompok dau gisom nokogonop di pointopot linaab id puyut do pampos. Poomitanan, montok iso ruhang Riemann poingkaa<ref>https://mathworld.wolfram.com/RiemannIntegral.html</ref>:

<math>L = \sum_{k=1}^n f(x^{*}_{k}) \Delta x_{k}, \quad D = \lim_{\max{\Delta x_{k}} \rightarrow 0} L</math>

<math>D = \int_{a}^{b} f(x) dx</math>

It gatang <math>\max{\Delta x_{k}}</math> roitan do ginayo gamit (''mesh size'') do pinalangod it sopo <math>[a, b]</math> kumaa ninivohit <math>\Delta x_{k}</math>.

== Bahazan ==
It sompuuvan nopo nga' kopio iso pananda otinulid (''linear operator'') toi ko' pamampos (''functor'') tu' osuanan om monihis do pampos. Hingkaa tongo bahazan pintukaon: <ref>https://tutorial.math.lamar.edu/classes/calci/ProofIntProp.aspx</ref>

=== Togom oili ===
<math>\int kf(x)\ dx = k \int f(x)\ dx</math>

=== Rinuhang duo toi lobi sompuuvan ===
<math>\int f(x) + g(x)\ dx = \int f(x)\ dx + \int g(x)\ dx</math>

=== Nupi gatang -1 om kobolikan do vohit ===
<math>- \int_{a}^{b} f(x)\ dx = \int_{b}^{a} f(x)\ dx</math>

== Sukuon ==
{{Reflist}}