Riemann Sum: Imwe yeMbiru dzeIntegral Calculus
Mumasvomhu, kunyanya mu integral calculus, pfungwa yeRiemann sum ine basa guru. Yakaunzwa nenyanzvi yemasvomhu yekuGermany ine mukurumbira Bernhard Riemann, nzira iyi inzira inokosha yekutsanangura kubatanidzwa kwebasa mukati menguva yakatarwa. Kunzwisisa Riemann sum kunotibvumira kufungidzira nzvimbo iri pasi pe curve, kushandiswa kwakakosha muminda yakawanda yesainzi netekinoroji, kubva kufizikisi kusvika kuhupfumi.
Kuti tinzwisise kukosha kweRiemann sum, tinofanira kuongorora zvinhu zvayo zvekutanga, kusanganisira kupatsanurana kwezvikamu, kuona mapoinzi ekuongorora, kuvaka ma sums, uye mashandisirwo awo mukubatanidzwa. Ngatinyatsoongororai nyaya iyi zvakadzama.
Nhanganyaya kuPfungwa Dzekutanga
Riemann sum inzira yekuverenga definite integral yebasa pamusoro pe closed interval \([a, b]\). Iyi nzira inosanganisira kupatsanura interval kuita subintervals diki, kuongorora basa panzvimbo dzakatarwa mu subinterval yega yega, uye wobva wabatanidza zvigadzirwa zve function values nehurefu hwe subintervals dzinoenderana.
Chikamu chepakati
Danho rekutanga pakutsanangura huwandu hweRiemann nderekupatsanura nguva \([a, b]\) kuita zvikamu zvidiki zvehurefu hwakapihwa. Ngatitii nguva \([a, b]\) yakakamurwa kuita \(n\) zvikamu zvakaenzana, zvino:
\[ \Delta x = \frac{b – a}{n} \]
Chikamu chimwe nechimwe chepakati chine urefu \(\Delta x\), uye idzi nzvimbo dzekuparadzanisa dzinowanzova \((x_0, x_1, x_2, …, x_n)\), uko \(x_0 = a\), \(x_1 = a + \Delta x\), \(x_2 = a + 2\Delta x\), zvichingodaro kusvika \(x_n = b\).
Kugadzwa kweNzvimbo yeKuongorora
Pachikamu chimwe nechimwe chepakati \([x_{i-1}, x_i]\), poindi yekuongorora \(x_i \) inodiwa iri mukati mechikamu ichocho chepakati. Poindi iyi inogona kutsanangurwa seinotevera:
1. Nzvimbo Yekuruboshwe: \(x_i^ = x_{i-1}\)
2. Nzvimbo Yekurudyi: \(x_i^ = x_i\)
3. Pakati: \(x_i^ = \frac{x_{i-1} + x_i}{2}\)
4. Mapoinzi Asina Kurongwa: Poinzi yega yega \(x_i \) ipoinzi isina kurongwa mu \([x_{i-1}, x_i]\)
Kusarudzwa kwemapoinzi ekuongorora kunogona kukanganisa mhedzisiro yeRiemann sum, kunyanya kana basa racho risingaenderere mberi kana kuti richichinja nekukurumidza.
Kuumbwa kweSum
Kana kupatsanurana kwepakati uye kuongorora mapoinzi kwapera, danho rinotevera nderekuverenga kukosha kwebasa panzvimbo yega yega yekuongorora \(f(x_i^ )\) uye kuwedzera kukosha ikoko nehukuru hwepakati penguva \(\Delta x\). Mari yeRiemann \(R\) inotsanangurwa se:
\[ R = \sum_{i=1}^nf(x_i^ ) \Delta x \]
Kana nhamba yezvikamu zvepakati \(n\) yawedzerwa pasina kusungwa (\(n \rightarrow \infty\)), kureba kwezvikamu zvepakati \(\Delta x\) kunova kudiki zvikuru uye huwandu hweRiemann hunosvika pakubatanidzwa kwebasa \(f\) pane chikamu \([a, b]\). Muganho uyu wakanyorwa seizvi:
\[ \int_a^bf(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^nf(x_i^ ) \Delta x \]
Muenzaniso weKushandiswa kweRiemann Sum
Semuenzaniso, ngatishandisei Riemann sum kuti tione integral yebasa \(f(x) = x^2\) pane interval \([0, 1]\).
Danho 1: Kupatsanurana Kwenguva
Ngatitii taparadzanisa interval \([0, 1]\) kuita \(n\) subintervals dzine urefu hwakaenzana, ipapo kureba kwe subintervals ndekwekuti:
\[ \Delta x = \frac{1 – 0}{n} = \frac{1}{n} \]
Danho rechipiri: Nzvimbo yekuongorora
Shandisa midpoint \(x_i \) kuongorora basa pane imwe neimwe subinterval \([x_{i-1}, x_i]\):
\[ x_i^ = \frac{x_{i-1} + x_i}{2} = \frac{\left(\frac{i-1}{n}\right) + \left(\frac{i}{n}\right)}{2} = \frac{2i – 1}{2n} \]
Danho rechitatu: Verenga Huwandu Hwese
Kukosha kwebasa \(f(x_i^ ) = \left( \frac{2i – 1}{2n} \right)^2 = \frac{(2i-1)^2}{4n^2}\), ipapo huwandu hweRiemann hunova:
\[ R = \sum_{i=1}^nf\left(\frac{2i – 1}{2n}\right) \Delta x = \sum_{i=1}^n \frac{(2i-1)^2}{4n^2} \cdot \frac{1}{n} = \frac{1}{4n^3} \sum_{i=1}^n (2i-1)^2 \]
Nekuongorora kwakawedzerwa, huwandu hwemasikweya enhamba dzisina kurongeka hunopa chiratidzo che sigma chinogona kurerutswa kusvika chasvika pamuganho.
Pakupedzisira, sezvo \(n\) ichienda ku infinity, kukosha kweRiemann sum kuchasvika pamhedzisiro ye exact integral:
\[ \lim_{n \to \infty} \frac{1}{4n^3} \sum_{i=1}^n (2i-1)^2 = \frac{1}{3} \]
Uye mumhedzisiro yekuongorora ye integral tinowana:
\[ \int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1}{3} \]
Mhando dzakasiyana-siyana uye mashandisirwo eRiemann Sums
Kunze kwekubatanidzwa kwechinyakare, huwandu hweRiemann hunewo dzimwe nzira dzakasiyana, dzinosanganisira huwandu hweRiemann-Kronecker uye huwandu hweRiemann-Stieltjes hwenzvimbo dzemetric uye mashandisirwo akafara mukuongorora mashandiro. Hunoumbawo hwaro hwenzira dzekuverenga dzakadai senzira dzeTrapzoid naSimpson dzinoshandiswa mukuverenga kwesainzi.
Kuvhara
Mari yeRiemann inopa nzira yakasimba uye inochinjika yekutsanangura nekuverenga zvinhu zvakasiyana-siyana mumasvomhu. Sechishandiso chekudzidzisa matambudziko makuru ekuverenga, kunzwisisa kwakakwana kwepfungwa iyi kunovhura ruzivo rwekushandiswa kwakakura kwezvinhu zvakasiyana-siyana muhupenyu chaihwo, mune zvese sainzi chaiyo uye munzvimbo dzezvehupfumi. Bernhard Riemann haana kungowedzera dzidziso yemasvomhu nekuwanikwa uku chete asiwo akavhura nzira itsva mukuongorora kwazvino kwezvikamu zvakasiyana.
