Kubatanidzwa
Ma-integrals ipfungwa huru mukuverenga, ine hukama hwakanyanya ne-derivatives. Kunyange zvazvo ma-derivatives achitsanangura shanduko mu-slope kana slope ye-curve panzvimbo yakatarwa, ma-integrals, kune rumwe rutivi, anosanganisira nzvimbo yese iri pasi pe-curve pamusoro pe-interval yakatarwa. Ma-integrals zvishandiso zvakakosha mumasvomhu, fizikisi, engineering, nedzimwe nzvimbo dzakawanda. Chinyorwa chino chichakurukura zvakadzama nezvepfungwa ye-integrals, kusanganisira nhoroondo yavo, tsananguro, mhando dze-integrals, mashandisirwo, uye mimwe mienzaniso yematambudziko e-integral.
Nhoroondo Yakabatana
MaIntegrals akatanga kuunzwa nemasayendisiti ekare echiGiriki kuburikidza nenzira yekuneta yakagadzirwa naEudoxus uye yakazofarirwa naArchimedes. Zvisinei, kufambira mberi kwemaIntegrals emazuva ano kunogona kuteverwa kusvika muzana remakore rechi17 nebasa raIsaac Newton naGottfried Wilhelm Leibniz. Newton naLeibniz, kunyangwe vaishanda vega, vakagadzira calculus yaisanganisira maIntegrals neDerivatives, pfungwa mbiri dzine chekuita neFundamental Theorem of Calculus. Iyi theorem inotaura kuti kubatanidzwa nekusiyanisa mabasa akasiyana.
Newton akashandisa ma integrals muchimiro chefizikisi kuti averenge nzvimbo iri pasi pe curve inomiririra shanduko yekukurumidza nekufamba kwenguva, pfungwa yakakosha kumakanika ekare. Kune rumwe rutivi, Leibniz akagadzira integral notation ∫ yatinoshandisa nhasi, yakabva pabhii refu "s," pfupiso ye summa (Latin rinoreva "sum").
Tsanangudzo yeIntegral
Kubatanidzwa kwebasa kunogona kufungidzirwa se "anti-derivative." Pamutemo, kune mhando mbiri dzekubatanidzwa dzinowanzoshandiswa: integral isingaverengeki uye integral yakavanzika.
Kubatana Kusingaperi
Chinhu chisingaperi chebasa f(x) ibasa F(x) rine derivative yaF(x) iri f(x). Mashoko aro akanyorwa seizvi:
∫ f(x) dx = F(x) + C
apo C i "constant" yekubatanidzwa iyo inodiwa nekuti "derivative" ye "constant" izero. Muenzaniso wakapfava we "integral" isingaverengeki ndewekuti:
∫ 2x dx = x² + C
Yakabatana Zvisingaperi
Chinhu chakakosha chebasa f(x) kubva kuna a kusvika kuna b chinotsanangura nzvimbo iri pasi pekongiri kubva kuna x = a kusvika kuna x = b. Yakanyorwa seizvi:
∫[a,b] f(x) dx = F(b) – F(a)
apo F(x) iri chinhu chinopesana nef(x). Semuenzaniso:
∫[1,3] 2x dx = (3² – 1²) = 8
Mhando dzeIntegrals
Pamusoro pezvinhu zvakakosha zvataurwa pamusoro apa, kune mamwe marudzi akasiyana-siyana ezvinhu zvinoshandiswa mumasvomhu uye mashandisirwo anoitwa. Mamwe acho ndeaya:
Yakabatanidzwa Kaviri
Chinhu chinobatanidza kaviri chinosanganisira kubatanidzwa kwezvinhu zviviri kana kupfuura. Chinowanzoshandiswa mukuverenga zvinhu zvakawanda kuti chiverengere mavhoriyamu ari pasi pevhu kana kuti nhamba dzemabasa ezvinhu zviviri kana kupfuura. Mienzaniso inosanganisira:
∫∫_R f(x,y) dA
Lebesgue Integral
Iyo Lebesgue integral inoreva huwandu hweRiemann integral inoshandiswa kubata mabasa asina kurongeka zvakanyanya. Iyi integral inonyanya kukosha mukuongorora chaiko uye dzidziso yeprobability.
Mutsetse Wakabatana
Mutsetse unobatanidza basa riri munzira yakatarwa mundege kana muchadenga. Unobatsira zvikuru mufizikisi neinjiniya pakuverenga basa rinoitwa nesimba riri munzira yakatarwa.
∫_C F · chiremba
Pamusoro Pechinhu Chinobatanidzwa
Chinhu chinosanganisa zvinhu zviviri (double integral) chinoshandura chinhu chinosanganisa zvinhu zviviri (double integral) kuti chishandiswe panzvimbo dziri munzvimbo dzine mativi matatu. Chinowanzo shandiswa mu electrodynamics uye fluid mechanics.
∫∫_S f(x,y,z) dS
Kushandiswa Kwakabatana
MaIntegrals ane mashandisirwo akasiyana-siyana muzvikamu zvakasiyana zvesainzi. Mamwe emashandisirwo akakosha anosanganisira:
Fizikisi
Mufizikisi, zvinhu zvidiki (integrals) zvinowanzo shandiswa kuverenga basa, simba, nezvimwe zvinhu zvakasiyana-siyana. Semuenzaniso, zvinhu zvidiki (integrals) zvinoshandiswa kuverenga basa rinoitwa nesimba riri mukufamba kwechinhu:
W = ∫ F(x) dx
Zvinhu zvinosanganisa zvinhu zvakakoshawo pakutsvaga huremu, pakati pehuremu, nguva yekusagadzikana, uye kupararira kwehuremu muzvinhu zvinoonekwa.
Statistics
Muma statistics ne probability theory, ma integrals anoshandiswa kuverenga probabilities netarisiro dzekugoverwa kwakasiyana-siyana kwe probability. Muenzaniso ndewekuverenga cumulative probability ye continuous random variable:
P(X ≤ x) = ∫_(-∞, x) f(t) dt
upfumi
Muzvehupfumi, zvinhu zvinosanganisa zvinoshandiswa kuverenga purofiti yese, mari inoshandiswa, uye mari inosara kubva kumutengi/mugadziri. Semuenzaniso, kuverenga mari yose inoshandiswa muchikamu chemari iri pasi pemutengo wepakati:
Mutengo Wose = ∫ MC(q) dq
zvekushandisa
Muinjiniya, zvinhu zvinosanganisa zvinhu zvinoshandiswa zvakanyanya (integrals) zvinoshandiswa zvakanyanya mukuongorora kushushikana, kuchinja kwesimba remagetsi (stress analysis), uye kutamisa kupisa (heat transfer). Kudzora kunoshandisawo zvinhu zvinosanganisa zvinhu zvinoshandiswa zvakanyanya (integrals) muzvinodzora zvePID (Proportional-Integral-Derivative).
Mienzaniso yeMatambudziko Akabatana
Heano mimwe mienzaniso yematambudziko akakosha uye mhinduro dzawo:
Muenzaniso 1: Kusaguma Kwakabatana
Tsvaga chinhu chisingaperi che f(x) = 3x²:
∫ 3x² dx
Solusi:
Iyi integral inogona kugadziriswa uchishandisa mutemo wekutanga we integral. Tinowedzera imwe ku exponent tobva tagovanisa ne exponent itsva:
∫ 3x² dx = 3 (1/3)x³ = x³ + C
Muenzaniso 2: Definite Integral
Tsvaga nzvimbo iri pasi pekakunguru f(x) = 2x kubva pa1 kusvika pa4:
∫[1,4] 2x dx
Solusi:
Kutanga, tinowana chinhu chinopesana ne2x, chinova x². Zvadaro, tinoshandisa miganhu yepasi neyepamusoro:
∫[1,4] 2x dx = [x²]₁^₄ = 4² – 1² = 16 – 1 = 15
Mhedziso
MaIntegrals ipfungwa dzakakosha mukuverenga, achibata pfungwa yenzvimbo iri pasi pemugero uye mashandisirwo ayo akawanda. Kubva munhoroondo yavo muGreece yekare kusvika kubudiriro huru yaNewton naLeibniz, kusvika kumarudzi akasiyana-siyana nemashandisirwo avo musainzi neinjiniya, maIntegrals anoita basa rakakosha muhurongwa hwemazuva ano hwemasvomhu. Kunzwisisa maIntegrals hakungotibatsiri chete kugadzirisa matambudziko emasvomhu asiwo kunotipa ruzivo rwemashandisirwo avo akawanda muhupenyu hwezuva nezuva uye munyika yehunyanzvi.
