Cov Lus Nug Piv Txwv Sib Tham Txog Kev Siv Integrals Hauv Kev Lag Luam thiab Kev Lag Luam
Pendahuluuan
Cov integrals yog lub tswv yim tseem ceeb hauv kev xam lej thiab muaj ntau daim ntawv thov hauv ntau qhov chaw, suav nrog kev lag luam thiab kev lag luam. Hauv cov ntsiab lus no, integrals feem ntau siv los tshuaj xyuas tag nrho cov nyiaj tau los, tus nqi, cov nyiaj tau los, thiab kev siv thiab kev tsim khoom. Kev nkag siab txog kev siv cov integrals hauv kev lag luam thiab kev lag luam tsis yog tsuas yog pab daws cov teeb meem kev siv tshuab xwb tab sis kuj muab kev nkag siab tob dua txog kev ua lag luam, kev txiav txim siab, thiab kev npaj tswv yim.
Cov Kev Siv Sib Txuas Lus hauv Kev Lag Luam thiab Kev Lag Luam
1. Xam Tag Nrho Cov Nyiaj Tau Los
Yuav xam tag nrho cov nyiaj tau los, peb feem ntau yuav tsum suav cov nyiaj tau los me me uas tau txais los ntawm kev muag cov khoom ib leeg. Yog tias tus nqi ntawm cov khoom sib txawv nyob ntawm qhov ntau npaum li cas muag, ces qhov kev ua haujlwm nqe-qhov ntau yuav tsum tau koom ua ke los txiav txim siab tag nrho cov nyiaj tau los.
Piv txwv teeb meem:
Xav tias tus nqi \(p\) ntawm ib yam khoom nyob ntawm seb muaj pes tsawg yam khoom \(q\) muag, uas yog muab los ntawm cov haujlwm hauv qab no:
\[ p(q) = 100 – 2q \]
Xam tag nrho cov nyiaj tau los yog tias muag tau 10 units khoom.
Kev daws teeb meem:
Tag nrho cov nyiaj tau los \ (R \) yog qhov sib xyaw ntawm tus nqi hla qhov ntau ntawm 0 txog \ (Q \) units.
\[ R = \int_{0}^{Q} p(q) \, dq \]
Nrog \( p(q) = 100 - 2q \) thiab \( Q = 10 \):
\[ R = \int_{0}^{10} (100 – 2q) \, dq \]
Yog li, peb xam qhov integral:
\[ R = \left[ 100q – q^2 \right]_{0}^{10} \]
Soj ntsuam cov kev txwv ntawm qhov sib xyaw ua ke:
\[ R = sab laug (100 cdot 10 – 10^2 \sab xis) – sab laug (100 cdot 0 – 0^2 \sab xis) \]
\[ = 1000 – 100 \]
\[ = 900 \]
Yog li, tag nrho cov nyiaj tau los yog muag tau 10 units yog 900.
2. Xam Tag Nrho Tus Nqi
Kev siv cov integrals hauv kev xam tag nrho cov nqi tsim khoom muaj txiaj ntsig zoo heev, tshwj xeeb tshaj yog thaum cov nqi ntxiv tsis ruaj khov thiab nyob ntawm qhov ntau npaum li cas tsim tawm. Cov nqi ntxiv tuaj yeem piav qhia tias yog cov derivative ntawm tag nrho cov nqi, thiab kom nrhiav tau tag nrho cov nqi peb yuav tsum tau integrate.
Piv txwv teeb meem:
Yog tias tus nqi ntxiv \(MC \) rau kev tsim cov khoom \(q \) units yog muab los ntawm:
\[ MC(q) = 50 + 3q^2 \]
Xam tag nrho tus nqi yog tias muaj 5 units ntawm cov khoom raug tsim tawm los ntawm kev xav tias tus nqi ruaj khov \(C \) ntawm 200.
Kev daws teeb meem:
Tag nrho tus nqi \( TC \) yog qhov sib xyaw ntawm tus nqi ntxiv rau tus nqi ruaj khov:
\[ TC = \int_{0}^{Q} MC(q) \, dq + C \]
Nrog \( MC(q) = 50 + 3q^2 \) thiab \( Q = 5 \):
\[ TC = \int_{0}^{5} (50 + 3q^2) \, dq + 200 \]
Peb xam qhov sib npaug:
\[ TC = \left[ 50q + q^3 \right]_{0}^{5} + 200 \]
Soj ntsuam cov kev txwv ntawm qhov sib xyaw ua ke:
\[ TC = \sab laug ( 50 \cdot 5 + 5^3 \sab xis) – \sab laug ( 50 \cdot 0 + 0^3 \sab xis) + 200 \]
\[ = \sab laug ( 250 + 125 \sab xis) + 200 \]
\[ = 375 + 200 \]
\[ = 575 \]
Yog li, tag nrho tus nqi los tsim 5 units ntawm cov khoom yog 575.
3. Xam Kev Siv Cov Khoom Siv
Cov lej sib xyaw kuj tseem siv los xam tag nrho cov kev siv lossis kev siv ntawm ib qho peev txheej dhau ib lub sijhawm. Qhov no yog qhov tseem ceeb tshaj plaws hauv cov ntsiab lus lag luam uas cuam tshuam nrog cov peev txheej xws li lub zog, cov ntaub ntawv, lossis tib neeg.
Piv txwv teeb meem:
Tus nqi siv hluav taws xob txhua hnub \(E \) hauv lub Hoobkas ua raws li cov haujlwm exponential hauv qab no:
\[ E(t) = 10e^{0.1t} \]
Xam tag nrho cov zog siv rau 10 hnub.
Kev daws teeb meem:
Tag nrho cov zog siv \(C \) dhau lub sijhawm [0, T] yog qhov sib xyaw ntawm cov nqi siv zog ntawd:
\[ C = \int_{0}^{T} E(t) \, dt \]
Nrog \( E(t) = 10e^{0.1t} \) thiab \( T = 10 \):
\[ C = \int_{0}^{10} 10e^{0.1t} \, dt \]
Yuav kom xam tau qhov integral, peb siv tau cov txheej txheem hloov chaw:
Cia \( u = 0.1t \), ces \( du = 0.1 \, dt \), los yog \( dt = \frac{du}{0.1} \),
\[C = \int_{0}^{1} 10e^{u} \frac{du}{0.1} \]
\[ = 100 \int_{0}^{1} e^{u} \, du \]
\[ = 100 \left[ e^{u} \right]_{0}^{1} \]
Soj ntsuam cov kev txwv ntawm qhov sib xyaw ua ke:
\[ C = 100 \sab laug ( e^{1} – e^{0} \sab xis) \]
\[ = 100 \sab laug ( e – 1 \sab xis) \]
Nrog \( e \approx 2.718 \):
\[ C \kwv yees li 100 (2.718 – 1) \]
\[ = 100 \times 1.718 \]
\[ = 171.8 \]
Yog li, tag nrho cov zog siv rau 10 hnub yog 171.8 units ntawm lub zog.
Xaus
Lub tswv yim ntawm kev sib xyaw ua ke yog qhov tseem ceeb hauv kev lag luam thiab kev lag luam, vim nws tso cai rau cov kws tshuaj ntsuam xyuas thiab cov neeg txiav txim siab los xam thiab kwv yees cov hloov pauv tseem ceeb xws li cov nyiaj tau los, cov nqi, thiab kev siv. Kev nkag siab txog yuav ua li cas siv cov kev sib xyaw ua ke hauv cov ntsiab lus sib txawv no tuaj yeem muab qhov zoo dua kev sib tw thiab kev nkag siab zoo dua rau kev ua haujlwm lag luam. Vam tias, cov teeb meem piv txwv no yuav pab koj nkag siab txog kev siv cov kev sib xyaw ua ke hauv kev lag luam thiab kev lag luam.
