Kev npam
Cov integrals yog ib lub tswv yim tseem ceeb hauv kev xam lej, uas muaj feem cuam tshuam nrog cov derivatives. Thaum cov derivatives piav qhia txog kev hloov pauv ntawm qhov nqes hav lossis qhov nqes hav ntawm ib qho nkhaus ntawm ib qho chaw tshwj xeeb, cov integrals, ntawm qhov tod tes, suav nrog tag nrho thaj chaw hauv qab qhov nkhaus dhau ib lub sijhawm tshwj xeeb. Cov integrals yog cov cuab yeej tseem ceeb hauv kev lej, physics, engineering, thiab ntau lwm yam. Tsab xov xwm no yuav tham txog lub tswv yim ntawm integrals kom ntxaws, suav nrog lawv keeb kwm, kev txhais, hom integrals, kev siv, thiab qee qhov piv txwv ntawm cov teeb meem integral.
Keeb Kwm Sib Koom Ua Ke
Cov integrals tau qhia thawj zaug los ntawm cov kws tshawb fawb Greek thaum ub los ntawm txoj kev exhaustion uas Eudoxus tsim thiab tom qab ntawd Archimedes tau nrov npe. Txawm li cas los xij, kev txhim kho ntawm cov integrals niaj hnub no tuaj yeem taug qab rov qab mus rau xyoo pua 17th nrog rau txoj haujlwm ntawm Isaac Newton thiab Gottfried Wilhelm Leibniz. Newton thiab Leibniz, txawm hais tias ua haujlwm ywj pheej, tau tsim cov calculus uas suav nrog integrals thiab derivatives, ob lub tswv yim cuam tshuam los ntawm Fundamental Theorem ntawm Calculus. Cov theorem no hais tias kev koom ua ke thiab kev sib txawv yog cov haujlwm rov qab.
Newton siv cov integrals hauv cov ntsiab lus ntawm physics los xam thaj tsam hauv qab ib daim kab uas sawv cev rau qhov kev hloov pauv ntawm qhov ceev nrog lub sijhawm, lub tswv yim tseem ceeb rau cov kws kho tshuab classical. Leibniz, ntawm qhov tod tes, tau tsim cov cim integral ∫ uas peb siv niaj hnub no, los ntawm tsab ntawv ntev "s," luv rau summa (Latin rau "sum").
Kev Txhais ntawm Kev Sib Koom Tes
Tus lej sib xyaw ntawm ib qho kev ua haujlwm tuaj yeem xav txog tias yog "anti-derivative." Feem ntau, muaj ob hom lej sib xyaw uas feem ntau siv: tus lej sib xyaw tsis paub meej thiab tus lej sib xyaw tsis paub meej.
Kev Sib Koom Tes Tsis Paub Kawg
Tus lej sib piv tsis kawg ntawm ib qho kev ua haujlwm f(x) yog qhov kev ua haujlwm F(x) uas nws cov derivative ntawm F(x) yog f(x). Nws cov cim sau ua:
∫ f(x) dx = F(x) + C
qhov twg C yog ib qho tsis hloov pauv ntawm kev sib koom ua ke uas xav tau vim tias qhov derivative ntawm ib qho tsis hloov pauv yog xoom. Ib qho piv txwv yooj yim ntawm ib qho kev sib koom ua ke tsis paub meej yog:
∫ 2x dx = x² + C
Kev Sib Koom Tes Tseeb
Tus lej integral tseeb ntawm ib qho function f(x) los ntawm a mus rau b piav qhia txog thaj tsam hauv qab qhov nkhaus los ntawm x = a mus rau x = b. Nws yog sau li no:
∫[a,b] f(x) dx = F(b) – F(a)
qhov twg F(x) yog tus antiderivative ntawm f(x). Piv txwv li:
∫[1,3] 2x dx = (3² – 1²) = 8
Hom Kev Sib Koom Tes
Ntxiv rau cov integrals yooj yim uas tau tham saum toj no, muaj ntau hom integrals siv rau hauv kev suav lej thiab kev siv ua haujlwm. Qee qhov ntawm cov no yog:
Ob Chav Sib Koom Ua Ke
Ib qho kev sib xyaw ua ke ob npaug suav nrog kev koom ua ke ntawm ob lossis ntau qhov hloov pauv. Nws feem ntau siv rau hauv kev suav ntau qhov ntev los xam cov ntim hauv av lossis cov nqi lej rau cov haujlwm ntawm ob lossis ntau qhov hloov pauv. Piv txwv li:
∫∫_R f(x,y) dA
Lebesgue Integral
Tus lej Lebesgue integral yog kev siv dav dav ntawm Riemann integral uas siv los tswj cov haujlwm tsis xwm yeem. Tus lej no tseem ceeb heev rau kev tshuaj xyuas tiag tiag thiab kev xav txog qhov muaj feem yuav tshwm sim.
Kab Sib Koom Ua Ke
Ib txoj kab sib xyaw ua ke ib qho kev ua haujlwm raws li ib txoj kab nkhaus lossis txoj kev hauv ib lub dav hlau lossis qhov chaw. Nws muaj txiaj ntsig zoo heev hauv physics thiab engineering rau kev xam cov haujlwm ua los ntawm lub zog raws li txoj kev.
∫_C F · dr
Kev Sib Koom Tes Nto
Qhov kev sib xyaw ua ke ntawm qhov chaw hloov kho qhov kev sib xyaw ua ke ob npaug kom siv rau cov chaw hauv qhov chaw peb-seem. Nws feem ntau siv rau hauv electrodynamics thiab fluid mechanics.
∫∫_S f(x,y,z) dS
Daim Ntawv Thov Sib Koom Tes
Cov integrals muaj ntau yam kev siv hauv ntau qhov chaw ntawm kev tshawb fawb. Qee qhov kev siv tseem ceeb tshaj plaws suav nrog:
Kev Kawm Txog Lub Cev
Hauv kev kawm txog physics, cov integrals feem ntau siv los xam cov haujlwm, lub zog, thiab ntau yam khoom siv. Piv txwv li, cov integrals siv los xam cov haujlwm uas lub zog ua hauv kev txav ntawm ib qho khoom me me:
W = ∫ F(x) dx
Cov integrals kuj tseem ceeb rau kev nrhiav qhov hnyav, qhov chaw ntawm qhov hnyav, lub sijhawm ntawm inertia, thiab kev faib tawm ntawm qhov hnyav hauv cov khoom siv lub cev.
Kev txheeb cais
Hauv kev suav lej thiab kev xav txog qhov muaj feem yuav tshwm sim, cov integrals siv los xam cov feem yuav tshwm sim thiab kev cia siab ntawm ntau yam kev faib tawm ntawm qhov muaj feem yuav tshwm sim. Ib qho piv txwv yog xam cov feem yuav tshwm sim ntawm cov hloov pauv random tas mus li:
P(X ≤ x) = ∫_(-∞, x) f(t) dt
Kev lag luam
Hauv kev lag luam, cov integrals siv los xam tag nrho cov nyiaj tau los, cov nqi, thiab cov nyiaj seem ntawm cov neeg siv khoom/tus tsim khoom. Piv txwv li, los xam tag nrho cov nqi hauv cov ntsiab lus ntawm cov nqi ntxiv:
Tag Nrho Tus Nqi = ∫ MC(q) dq
kev
Hauv kev tsim kho vaj tse, cov integrals feem ntau siv rau hauv kev tshuaj xyuas kev ntxhov siab, kev hloov pauv, thiab kev hloov pauv cua sov. Kev tswj hwm engineering kuj siv cov integrals rau cov kab ke tswj hwm hauv PID (Proportional-Integral-Derivative) controllers.
Piv txwv teeb meem sib xyaw
Nov yog qee cov piv txwv ntawm cov teeb meem tseem ceeb thiab lawv cov kev daws teeb meem:
Piv txwv 1: Kev Sib Koom Ua Ib Ke Tsis Paub Kawg
Nrhiav qhov indefinite integral ntawm f(x) = 3x²:
∫ 3x² dx
Solusi:
Qhov kev suav ua ke no siv tau txoj cai suav ua ke yooj yim. Peb ntxiv ib qho rau tus lej qhia thiab tom qab ntawd faib los ntawm tus lej qhia tshiab:
∫ 3x² dx = 3 (1/3)x³ = x³ + C
Piv txwv 2: Kev Sib Koom Ua Ke Tseeb
Nrhiav thaj tsam hauv qab nkhaus f(x) = 2x ntawm 1 txog 4:
∫[1,4] 2x dx
Solusi:
Ua ntej, peb nrhiav tau qhov antiderivative ntawm 2x, uas yog x². Tom qab ntawd, peb siv cov kev txwv qis dua thiab siab dua:
∫[1,4] 2x dx = [x²]₁^₄ = 4² – 1² = 16 – 1 = 15
Xaus
Cov integrals yog cov tswv yim tseem ceeb hauv calculus, ntes lub tswv yim ntawm thaj chaw hauv qab ib qho nkhaus thiab nws cov ntawv thov ntau yam. Txij li lawv keeb kwm hauv tebchaws Greece thaum ub mus rau kev txhim kho tseem ceeb los ntawm Newton thiab Leibniz, mus rau lawv ntau hom thiab daim ntawv thov hauv kev tshawb fawb thiab kev tsim kho, integrals ua lub luag haujlwm tseem ceeb hauv cov qauv lej niaj hnub. Kev nkag siab txog integrals tsis yog tsuas yog pab peb daws cov teeb meem lej xwb tab sis kuj qhib kev nkag siab rau lawv cov ntawv thov ntau yam hauv lub neej txhua hnub thiab lub ntiaj teb kev ua haujlwm.
