I-Integral
Ama-Integral angumqondo oyisisekelo ekubaleni, ahlobene kakhulu nama-derivatives. Ngenkathi ama-derivatives echaza ushintsho emthambekeni noma emthambekeni wejika endaweni ethile, ama-integral, ngakolunye uhlangothi, ahlanganisa yonke indawo ngaphansi komthambeke phezu kwesikhawu esithile. Ama-Integral angamathuluzi abalulekile kwizibalo, i-physics, ubunjiniyela, kanye neminye imikhakha eminingi. Lesi sihloko sizoxoxa ngomqondo wama-integral ngokuningiliziwe, okuhlanganisa umlando wawo, incazelo, izinhlobo zama-integral, izinhlelo zokusebenza, kanye nezinye izibonelo zezinkinga ezihlanganisiwe.
Umlando Ohlanganisiwe
Ama-Integral aqala ukwethulwa ososayensi basendulo baseGrisi ngendlela yokukhathala eyasungulwa ngu-Eudoxus futhi kamuva yathandwa ngu-Archimedes. Kodwa-ke, ukuthuthukiswa kwama-integral anamuhla kungalandelelwa emuva ekhulwini le-17 ngomsebenzi ka-Isaac Newton noGottfried Wilhelm Leibniz. UNewton noLeibniz, nakuba babesebenza ngokuzimela, bathuthukise i-calculus eyayihlanganisa ama-integral nama-derivatives, imiqondo emibili ehlobene ne-Fundamental Theorem of Calculus. Le theorem ithi ukuhlanganiswa nokwehlukaniswa kuyimisebenzi ephambene.
UNewton wasebenzisa ama-integral kumongo wefiziksi ukuze abale indawo ngaphansi kwejika elimele ushintsho lwejubane ngokuhamba kwesikhathi, umqondo obalulekile kuma-mechanics akudala. Ngakolunye uhlangothi, uLeibniz wathuthukisa i-integral notation ∫ esiyisebenzisayo namuhla, ethathwe ohlamvini olude “s,” olufushane lwe-summa (isiLatini esisho “isamba”).
Incazelo ye-Integral
Ukuhlanganiswa komsebenzi kungacatshangwa ngokuthi “i-anti-derivative.” Ngokwesiko, kunezinhlobo ezimbili zama-integral ezivame ukusetshenziswa: i-integral enganqunyelwe kanye ne-integral eqondile.
Okuhlanganisiwe Okungapheli
I-integral engapheli yomsebenzi f(x) umsebenzi F(x) osuselwa ku-F(x) ongu-f(x). Umbhalo wawo ubhalwe kanje:
∫ f(x) dx = F(x) + C
lapho u-C eyisiguquli sokuhlanganiswa esidingekayo ngoba i-derivative yesiguquli esingu-zero. Isibonelo esilula se-integral esingapheli yilesi:
∫ 2x dx = x² + C
Okuhlanganisiwe Okuqinisekile
I-integral eqondile yomsebenzi f(x) kusukela ku-a kuya ku-b ichaza indawo engaphansi kwejika kusukela ku-x = a kuya ku-x = b. Ibhalwe kanje:
∫[a,b] f(x) dx = F(b) – F(a)
lapho u-F(x) eyi-antiderivative ye-f(x). Isibonelo:
∫[1,3] 2x dx = (3² – 1²) = 8
Izinhlobo Zokuhlanganiswa
Ngaphezu kwezinto eziyisisekelo ezihlanganisiwe okuxoxwe ngazo ngenhla, kunezinye izinhlobo ezahlukene zezinto ezihlanganisiwe ezisetshenziswa ezibalweni nasekusetshenzisweni okusebenzayo. Ezinye zazo yilezi:
Okuhlanganisiwe Okubili
I-double integral ihilela ukuhlanganisa iziguquguquki ezimbili noma ngaphezulu. Ivame ukusetshenziswa ekubalweni kwe-multidimensional ukubala amavolumu angaphansi komhlaba noma amanani ezinombolo emisebenzini yeziguquguquki ezimbili noma ngaphezulu. Izibonelo zifaka:
∫∫_R f(x,y) dA
I-Lebesgue Integral
I-Lebesgue integral iwukujwayelekile kwe-Riemann integral esetshenziselwa ukuphatha imisebenzi engajwayelekile kakhulu. Le integral ibaluleke kakhulu ekuhlaziyeni kwangempela kanye nethiyori yamathuba.
Ukuhlanganiswa Komugqa
I-integral yomugqa ihlanganisa umsebenzi eceleni kwejika noma indlela ethile endizeni noma esikhaleni. Iwusizo kakhulu ku-physics kanye nobunjiniyela ekubaleni umsebenzi owenziwe amandla endleleni ethile.
∫_C F · uDkt.
Okuhlanganisiwe Komphezulu
I-surface integral iguqula i-double integral ukuze isetshenziswe ezindaweni ezisendaweni enezinhlangothi ezintathu. Ivame ukusetshenziswa kuma-electrodynamics kanye ne-fluid mechanics.
∫∫_S f(x,y,z) dS
Isicelo Esihlanganisiwe
Ama-Integral anezinhlobo eziningi zezicelo emikhakheni eyahlukene yesayensi. Ezinye zezicelo ezibaluleke kakhulu zifaka:
Ifiziksi
Ku-physics, ama-integral avame ukusetshenziswa ukubala umsebenzi, amandla, kanye nezinye izinto eziningi ezibonakalayo. Isibonelo, ama-integral asetshenziswa ukubala umsebenzi owenziwa amandla ekunyakazeni kwenhlayiya:
W = ∫ F(x) dx
Ama-Integrals nawo abalulekile ekutholeni isisindo, isikhungo sesisindo, umzuzu wokungahlali kahle, kanye nokusatshalaliswa kwesisindo ezintweni ezibonakalayo.
Izibalo
Ku-izibalo kanye nethiyori yamathuba, ama-integrals asetshenziswa ukubala amathuba kanye nokulindela kokusatshalaliswa kwamathuba ahlukahlukene. Isibonelo ukubala amathuba ahlanganisiwe e-continuous random variable:
I-P(X ≤ x) = ∫_(-∞, x) f(t) dt
umnotho
Kwezomnotho, ama-integrals asetshenziswa ukubala inzuzo iyonke, izindleko, kanye nemali eningi etholakala kubathengi/kubakhiqizi. Isibonelo, ukubala izindleko eziphelele ngokomongo wezindleko ezingaphansi:
Izindleko Eziphelele = ∫ MC(q) dq
lobuchwepheshe
Kubunjiniyela, ama-integral avame ukusetshenziswa ekuhlaziyeni ukucindezeleka, ukuguqulwa, kanye nokudluliselwa kokushisa. Ubunjiniyela bokulawula busebenzisa futhi ama-integral ezinhlelweni zokulawula ezihlanganisiwe kubalawuli be-PID (Proportional-Integral-Derivative).
Izibonelo Zezinkinga Ezihlanganisiwe
Nazi ezinye zezibonelo zezinkinga eziphelele kanye nezixazululo zazo:
Isibonelo 1: Okuhlanganisiwe Okungapheli
Thola i-integral engapheli ye-f(x) = 3x²:
∫ 3x² dx
Solusi:
Lokhu okubalulekile kungaxazululwa kusetshenziswa umthetho oyisisekelo wokubalulekile. Sifaka eyodwa ku-exponent bese sihlukanisa nge-exponent entsha:
∫ 3x² dx = 3 (1/3)x³ = x³ + C
Isibonelo 2: I-Definite Integral
Thola indawo engaphansi kwejika f(x) = 2x kusukela ku-1 kuya ku-4:
∫[1,4] 2x dx
Solusi:
Okokuqala, sithola i-antiderivative ye-2x, okungu-x². Bese, sisebenzisa imikhawulo ephansi nephezulu:
∫[1,4] 2x dx = [x²]₁^₄ = 4² – 1² = 16 – 1 = 15
Isiphetho
Ama-integral ayimiqondo ebalulekile ekubaleni, athatha umqondo wendawo engaphansi kwejika kanye nokusetshenziswa kwayo okuningi. Kusukela emlandweni wawo eGrisi lasendulo kuya ekuthuthukisweni okubalulekile kukaNewton noLeibniz, kuya ezinhlotsheni zawo ezahlukene kanye nokusetshenziswa kwawo kwesayensi nobunjiniyela, ama-integral adlala indima ebalulekile ohlakeni lwezibalo lwanamuhla. Ukuqonda ama-integral akusisizi nje kuphela ukuxazulula izinkinga zezibalo kodwa futhi kuvula ukuqonda ngezicelo zawo eziningi empilweni yansuku zonke kanye nasezweni lochwepheshe.
