Riemann Sum: E 'ngoe ea Litšiea tsa Calculus e Integral
Lipalong, haholo-holo ho integral calculus, khopolo ea kakaretso ea Riemann e bapala karolo ea bohlokoa. E hlahisitsoe ke setsebi sa lipalo sa Mojeremane se tummeng Bernhard Riemann, mokhoa ona ke tsela ea bohlokoa ea ho hlalosa kopanyo ea mosebetsi nakong ea nako e fanoeng. Ho utloisisa kakaretso ea Riemann ho re lumella ho hakanya sebaka se tlas'a mothapo, ts'ebeliso ea bohlokoa mafapheng a mangata a saense le theknoloji, ho tloha fisiks ho ea moruong.
Ho utloisisa bohlokoa ba kakaretso ea Riemann, re tlameha ho hlahloba likarolo tsa eona tsa motheo, ho kenyeletsoa karohano ea nako, ho fumana lintlha tsa tlhahlobo, ho haha likhakanyo, le ts'ebeliso ea tsona kopanyong. A re hlahlobeng taba ena ka botebo.
Selelekela sa Mehopolo ea Motheo
Kakaretso ea Riemann ke mokhoa oa ho bala karolo e tobileng ea mosebetsi holim'a karohano e koetsoeng \([a, b]\). Mokhoa ona o kenyelletsa ho arola karohano ka li-subinterval tse nyane, ho lekola mosebetsi lintlheng tse itseng ho subinterval ka 'ngoe, ebe ho akaretsoa lihlahisoa tsa boleng ba mosebetsi ka bolelele ba li-subinterval tse tsamaellanang.
Karolo ea karohano
Mohato oa pele oa ho hlalosa kakaretso ea Riemann ke ho arola karohano \([a, b]\) ka li-subinterval tsa bolelele bo fanoeng. A re re karohano \([a, b]\) e arotsoe ka likarolo tse lekanang \(n\), ebe:
\[ \Delta x = \frac{b – a}{n} \]
Karolo e 'ngoe le e 'ngoe e ka tlase e na le bolelele ba \(\Delta x\), 'me lintlha tsena tsa karohano hangata ke \((x_0, x_1, x_2, …, x_n)\), moo \(x_0 = a\), \(x_1 = a + \Delta x\), \(x_2 = a + 2\Delta x\), joalo-joalo ho fihlela ho \(x_n = b\).
Qeto ea Ntlha ea Tlhahlobo
Bakeng sa subinterval e 'ngoe le e 'ngoe \([x_{i-1}, x_i]\), ntlha ea tlhahlobo \(x_i \) ea hlokahala e ka hare ho subinterval eo. Ntlha ena e ka fumanoa ka tsela e latelang:
1. Ntlha e Letshehadi: \(x_i^ = x_{i-1}\)
2. Ntlha e ka ho le letona: \(x_i^ = x_i\)
3. Bohareng: \(x_i^ = \frac{x_{i-1} + x_i}{2}\)
4. Lintlha tse sa Lebelloang: \(x_i \) e 'ngoe le e 'ngoe ke ntlha e sa Lebelloang ho \([x_{i-1}, x_i]\)
Khetho ea lintlha tsa tlhahlobo e ka ama sephetho sa kakaretso ea Riemann, haholo-holo haeba mosebetsi o sa tsoele pele kapa o na le phetoho e potlakileng.
Sebopeho sa Kakaretso
Hang ha karohano ea karohano le ntlha ea tlhahlobo li phethetsoe, mohato o latelang ke ho bala boleng ba ts'ebetso ntlheng e 'ngoe le e 'ngoe ea tlhahlobo \(f(x_i^ )\) le ho atisa boleng boo ka bolelele ba subinterval \(\Delta x\). Kakaretso ea Riemann \(R\) e hlalosoa e le:
\[ R = \sum_{i=1}^nf(x_i^ ) \Delta x \]
Ha palo ea li-subintervals \(n\) e eketseha ntle le ho tlangwa (\(n \rightarrow \infty\)), bolelele ba subinterval \(\Delta x\) bo ba bonyenyane ka ho sa feleng mme kakaretso ea Riemann e atamela karolo e kopaneng ea ts'ebetso \(f\) karolong e pakeng \([a, b]\). Mongolo ona oa moeli o ngotsoe tjena:
\[ \int_a^bf(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^nf(x_i^ ) \Delta x \]
Mohlala oa ho kenya tšebetsong Riemann Sum
Jwalo ka mohlala, ha re sebediseng kakaretso ya Riemann ho fumana karolo e kopaneng ya mosebetsi \(f(x) = x^2\) karolong e pakeng \([0, 1]\).
Mohato oa 1: Karohano ea Nako
A re re re arola karohano \([0, 1]\) ka di-subinterval tsa \(n\) tsa bolelele bo lekanang, ebe bolelele ba di-subinterval ke:
\[ \Delta x = \frac{1 – 0}{n} = \frac{1}{n} \]
Mohato oa 2: Ntlha ea Tlhahlobo
Sebelisa ntlha e bohareng \(x_i \) ho lekola ts'ebetso ho subinterval e 'ngoe le e 'ngoe \([x_{i-1}, x_i]\):
\[ x_i^ = \frac{x_{i-1} + x_i}{2} = \frac{\left(\frac{i-1}{n}\right) + \left(\frac{i}{n}\right)}{2} = \frac{2i – 1}{2n} \]
Mohato oa 3: Bala Kakaretso
Boleng ba mosebetsi \(f(x_i^ ) = \left( \frac{2i – 1}{2n} \right)^2 = \frac{(2i-1)^2}{4n^2}\), ebe kakaretso ya Riemann e ba:
\[ R = \sum_{i=1}^nf\left(\frac{2i – 1}{2n}\right) \Delta x = \sum_{i=1}^n \frac{(2i-1)^2}{4n^2} \cdot \frac{1}{n} = \frac{1}{4n^3} \sum_{i=1}^n (2i-1)^2 \]
Ka tlhahlobo e 'ngoe, kakaretso ea lisekoere tsa linomoro tse sa tloaelehang e fana ka letšoao la sigma le ka nolofatsoang haholoanyane ho fihlela le fihla moeling.
Qetellong, ha \(n\) e ya ho infinity, boleng ba kakaretso ya Riemann bo tla atamela sephetho sa equilibrium e nepahetseng:
\[ \lim_{n \to \infty} \frac{1}{4n^3} \sum_{i=1}^n (2i-1)^2 = \frac{1}{3} \]
'Me sephethong sa tlhahlobo sa karolo e kopaneng re fumana:
\[ \int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1}{3} \]
Mefuta e fapaneng le Ts'ebeliso ea Riemann Sums
Ntle le kopanyo ya setso, kakaretso ya Riemann e boetse e na le diphetoho tse ding, ho kenyeletswa kakaretso ya Riemann-Kronecker le kakaretso ya Riemann-Stieltjes bakeng sa dibaka tsa metric le ditshebediso tse pharaletseng tlhahlobong ya tshebetso. E boetse e theha motheo wa mekgwa ya dipalo jwalo ka mekgwa ya Trapzoid le Simpson e sebediswang ho k'homphieutha ya mahlale.
Ho koala
Lipalo tsa Riemann li fana ka mokhoa o matla le o tenyetsehang oa ho hlalosa le ho bala likarolo tse kopaneng maemong a fapaneng a lipalo. Jwalo ka sesebelisoa sa ho ruta mathata a mantlha a kopaneng ho calculus, kutlwisiso e batsi ea mohopolo ona e bula temohisiso mabapi le ts'ebeliso e pharaletseng ea likarolo tse kopaneng bophelong ba 'nete, ka bobeli mahlaleng a nepahetseng le libakeng tsa moruo oa sechaba. Bernhard Riemann ha aa ka a ntlafatsa khopolo-taba ea lipalo feela ka tšibollo ena empa hape o butse litsela tse ncha tlhahlobong ea sejoale-joale ea likarolo tse kopaneng.
