Mehlala ea Lipotso Tse Buisanang ka Tšebeliso ea Li-Integrals Lefapheng la Moruo le Khoebo
Pendahuluan
Li-integral ke mohopolo oa bohlokoa ho calculus 'me li na le lits'ebetso tse ngata mafapheng a fapaneng, ho kenyeletsoa moruo le khoebo. Moelelong ona, li-integral hangata li sebelisoa ho sekaseka phaello eohle, litšenyehelo, chelete e kenang, le ts'ebeliso le mesebetsi ea tlhahiso. Ho utloisisa ts'ebeliso ea li-integral moruong le khoebong ha ho thuse feela ho rarolla mathata a tekheniki empa hape ho fana ka temohisiso e tebileng mabapi le matla a 'maraka, ho etsa liqeto le moralo oa maano.
Likopo tse Kopaneng tsa Moruo le Khoebo
1. Bala Kakaretso ea Lekeno
Ho bala kakaretso ea chelete e kenang, hangata re hloka ho kopanya chelete e kenang e nyane e fumanoeng thekisong ea li-unit ka bomong tsa sehlahisoa. Haeba theko ea sehlahisoa e fapana ho latela bongata bo rekisitsoeng, joale mosebetsi oa theko le bongata o tlameha ho kopanngoa ho fumana kakaretso ea chelete e kenang.
Mohlala oa mathata:
A re re theko \( p \) ea thepa e itšetlehile ka bongata ba sehlahisoa \( q \) se rekisitsoeng, bo fanoang ke mosebetsi o latelang:
\[ p(q) = 100 – 2q \]
Bala kakaretso ea chelete e kenang haeba ho rekisitsoe liyuniti tse 10 tsa thepa.
Tharollo:
Kakaretso ea chelete e kenang \(R \) ke karolo ea theko holim'a bongata ho tloha ho liyuniti tse 0 ho isa ho \(Q \).
\[ R = \int_{0}^{Q} p(q) \, dq \]
Ka \( p(q) = 100 – 2q \) le \( Q = 10 \):
\[ R = \int_{0}^{10} (100 – 2q) \, dq \]
Kahoo, re bala karolo e kopaneng:
\[ R = \left[ 100q – q^2 \right]_{0}^{10} \]
Lekola meeli ea karolo e kopaneng:
\[ R = \left( 100 \cdot 10 – 10^2 \right) – \left( 100 \cdot 0 – 0^2 \right) \]
\[ = 1000 – 100 \]
\[ = 900 \]
Kahoo, kakaretso ea chelete e kenang haeba li-unit tse 10 tsa thepa li rekisoa ke 900.
2. Bala Kakaretso ea Litšenyehelo
Tšebeliso ea li-integral ha ho baloa litšenyehelo tsohle tsa tlhahiso e molemo haholo, haholo-holo ha litšenyehelo tse ka thoko li sa fetohe 'me li itšetlehile ka bongata bo hlahisoang. Litšenyehelo tse ka thoko li ka hlalosoa e le derivative ea litšenyehelo tsohle, 'me ho fumana litšenyehelo tsohle re hloka ho li kopanya.
Mohlala oa mathata:
Haeba litšenyehelo tse ka thoko tsa ho hlahisa diyuniti tsa thepa di fanwa ke:
\[ MC(q) = 50 + 3q^2 \]
Bala kakaretso ea litšenyehelo haeba liyuniti tse 5 tsa thepa li hlahisoa ha ho nahanoa ka litšenyehelo tse tsitsitseng \(C \) tsa 200.
Tharollo:
Kakaretso ea litšenyehelo ke karolo ea bohlokoa ea litšenyehelo tse ka thoko hammoho le litšenyehelo tse tsitsitseng:
\[ TC = \int_{0}^{Q} MC(q) \, dq + C \]
Ka \( MC(q) = 50 + 3q^2 \) le \( Q = 5 \):
\[ TC = \int_{0}^{5} (50 + 3q^2) \, dq + 200 \]
Re bala karolo e kopaneng:
\[ TC = \left[ 50q + q^3 \right]_{0}^{5} + 200 \]
Lekola meeli ea karolo e kopaneng:
\[ TC = \left( 50 \cdot 5 + 5^3 \right) – \left( 50 \cdot 0 + 0^3 \right) + 200 \]
\[ = \left( 250 + 125 \left) + 200 \]
\[ = 375 + 200 \]
\[ = 575 \]
Kahoo, litšenyehelo tsohle tsa ho hlahisa liyuniti tse 5 tsa thepa ke 575.
3. Ho Bala Tšebeliso ea Lisebelisoa
Li-integral li boetse li sebelisoa ho bala kakaretso ea tšebeliso kapa tšebeliso ea mohloli ka nako e itseng. Sena se bohlokoa haholo maemong a khoebo a kenyeletsang mehloli e kang matla, thepa, kapa batho.
Mohlala oa mathata:
Sekhahla sa tshebediso ya eneji sa letsatsi le letsatsi \(E \) fekthering se latela mosebetsi o latelang wa exponential:
\[ E(t) = 10e^{0.1t} \]
Bala kakaretso ea tšebeliso ea matla ka matsatsi a 10.
Tharollo:
Kakaretso ea tšebeliso ea eneji \(C \) nakong ena [0, T] ke karolo ea bohlokoa ea sekhahla seo sa tšebeliso ea eneji:
\[ C = \int_{0}^{T} E(t) \, dt \]
Ka \( E(t) = 10e^{0.1t} \) le \( T = 10 \):
\[ C = \int_{0}^{10} 10e^{0.1t} \, dt \]
Ho bala karolo e kopaneng, re ka sebelisa mokhoa oa ho nkela sebaka:
A re \( u = 0.1t \), ebe \( du = 0.1 \, dt \), kapa \( dt = \frac{du}{0.1} \),
\[ C = \int_{0}^{1} 10e^{u} \frac{du}{0.1} \]
\[ = 100 \int_{0}^{1} e^{u} \, du \]
\[ = 100 \ka ho le letshehadi[ e^{u} \ka ho le letona]_{0}^{1} \]
Lekola meeli ea karolo e kopaneng:
\[ C = 100 \ka ho le letshehadi( e^{1} – e^{0} \ka ho le letona) \]
\[ = 100 \ka ho le letshehadi( e – 1 \ka ho le letona) \]
Ka \( e \hoo e ka bang 2.718 \):
\[ C \hoo e ka bang 100 (2.718 - 1) \]
\[ = 100 \makhetlo a 1.718 \]
\[ = 171.8 \]
Kahoo, tšebeliso eohle ea matla ka matsatsi a 10 ke li-unit tse 171.8 tsa matla.
Qetello
Khopolo ea li-integral e bohlokoa moruong le khoebong, kaha e lumella bahlahlobisisi le baetsi ba liqeto ho bala le ho bolela esale pele liphetoho tsa bohlokoa tse kang lekeno, litšenyehelo le tšebeliso. Ho utloisisa mokhoa oa ho sebelisa li-integral maemong ana a fapaneng ho ka fana ka monyetla oa tlholisano le temohisiso e betere mabapi le ts'ebetso ea khoebo. Re tšepa hore mehlala ena ea mathata e tla u thusa ho utloisisa lits'ebetso tse sebetsang tsa li-integral moruong le khoebong.
