Kopo e Kopanetsoeng
Li-integral ke mohopolo oa motheo lipalo, haholo-holo calculus. Li-integral li bapala karolo ea bohlokoa mafapheng a fapaneng a saense le theknoloji, ho kenyeletsoa fisiks, boenjiniere, moruo, baeloji, le tse ling. Sehloohong sena, re tla hlahloba ts'ebeliso ea li-integral maemong a fapaneng, a khopolo-taba le a sebetsang. Litšebeliso tsa integral li ka aroloa ka mekhahlelo e mengata e pharaletseng, joalo ka ho fumana sebaka, ho bala bophahamo ba modumo, tlhahlobo ea moruo, ho etsa mohlala oa 'mele le moralo oa boenjiniere.
1. Ho Fumana Sebaka sa Sebaka
E 'ngoe ea lits'ebetso tse tsebahalang haholo tsa li-integral ke ho fumana sebaka se ka tlas'a kotopo ea mosebetsi o fanoeng. Mohlala, haeba re na le mosebetsi \( f(x) \), sebaka se moeling oa kotopo pakeng tsa lintlha tse peli \(a\) le \(b\) ho x-axis se ka fumanoa ho sebelisoa motsoako o latelang:
\[ \mongolo{Sebaka} = \int_{a}^{b} f(x)\, dx \]
Mohlala, nahana ka mosebetsi o bonolo oa mola \( f(x) = 2x \). Ho fumana sebaka se ka tlas'a mothapo ho tloha \( x = 0 \) ho ea ho \( x = 3 \):
\[ \text{Area} = \int_{0}^{3} 2x\, dx = \left[ x^2 \right]_{0}^{3} = 3^2 – 0^2 = 9 \]
Sebaka sa sebaka seo ke diyuniti tse 9 tsa sebaka.
2. Ho Bala Bophahamo ba Molumo
Ntle le ho fumana sebaka sa sebaka, metsoako e ka boela ea sebelisoa ho bala bophahamo ba ntho e lekantsoeng ke kobeho kapa bokaholimo. Mekhoa e tsebahalang ea ho bala bophahamo e kenyelletsa mokhoa oa disc le mokhoa oa silindara.
2.1 Mokhoa oa Disc
Mokhoa oa disk o sebelisoa ho bala bophahamo ba ntho e tiileng e fumanoeng ka ho potoloha mothapo ho potoloha mothapo o le mong. Mohlala, bophahamo ba ntho bo fumanoeng ka ho potoloha mothapo \( y = f(x) \) ho potoloha mothapo oa x ho tloha \( x = a \) ho ea ho \( x = b \) ke:
\[ \text{Volume} = \pi \int_{a}^{b} \left( f(x) \right)^2\, dx \]
Mohlala, ho fumana bophahamo ba modumo bo fumanweng ka ho potoloha mothapo \( y = \sqrt{x} \) ho tloha \( x = 0 \) ho ya ho \( x = 2 \):
\[ \text{Volume} = \pi \int_{0}^{2} (\sqrt{x})^2\, dx = \pi \int_{0}^{2} x\, dx = \pi \left[ \frac{x^2}{2} \right]_{0}^{2} = \pi \left( \frac{4}{2} – 0 \right) = 2\pi \]
Mokhoa oa Silindara oa 2.2
Mokhoa oa silindara o sebelisoa ho bala bophahamo ba ntho e tiileng ka ho potoloha sekhutlo ho potoloha mothapo oa y. Ho sebelisoa mohopolo oa khoele e rapameng (axial):
\[ \text{Volume} = 2 \pi \int_{a}^{b} x \cdot f(x)\, dx \]
Mohlala, ho bala bophahamo ba modumo bo fumanweng ka ho potoloha mothapo \( y = x^2 \) ho tloha \( x = 0 \) ho ya ho \( x = 1 \) ho potoloha mothapo wa y:
\[ \text{Volume} = 2 \pi \int_{0}^{1} x \cdot x^2\, dx = 2 \pi \int_{0}^{1} x^3\, dx = 2 \pi \left[ \frac{x^4}{4} \right]_{0}^{1} = 2 \pi \left( \frac{1}{4} – 0 \right) = \frac{\pi}{2} \]
3. Tlhahlobo ea Moruo
Moruong, metsoako e kopaneng e sebelisetsoa merero e fapaneng, joalo ka ho bala keketseho ea mohlahisi le ea bareki le ho bolela esale pele kholo ea moruo. Mohlala, keketseho ea bareki e ka baloa ho sebelisoa metsoako e kopaneng ho fumana phapang pakeng tsa seo bareki ba ikemiselitseng ho se lefa le seo ba hlileng ba se lefang.
Mohlala, haeba mosebetsi wa tlhokeho \( p(x) \) o bontsha theko eo bareki ba ikemiseditseng ho e lefa bakeng sa diyuniti \( x \) tsa thepa, mme \( p_0 \) ke theko ya mmaraka, tjhelete e setseng ya bareki ho tloha ho 0 ho isa ho \( x_0 \) ke:
\[ \text{Bareki ba eketsehileng} = \int_{0}^{x_0} p(x)\, dx – p_0 \times x_0 \]
Mohlala o mong ke ho bala boleng ba hona joale ba phallo ea chelete ea nakong e tlang ka ho sebelisa mohopolo oa theolelo. Haeba phallo ea chelete ea nakong e tlang \( C(t) \) e lula e theoleloa ka sekhahla sa theolelo \( r \), boleng ba hona joale \( PV \) ke:
\[ PV = \int_{0}^{T} C(t) e^{-rt}\, dt \]
4. Ho Etsa Mehlala ea Fisiks
Li-integral li bapala karolo ea bohlokoa fisiks, li sebelisoa ho hlalosa melao e fapaneng ea fisiks le ho ntšetsa pele tlhahlobo ea litsamaiso tse fetohang.
4.1 Melao ea Tsamaiso
Mohlala, fisiks ea khale, melao ea Newton ea motsamao e ka hlalosoa ka sebopeho sa bohlokoa. Sebaka sa ntho e le mosebetsi oa nako se ka fumanoa ka ho kopanya lebelo la eona:
\[ x(t) = x(0) + \int_{0}^{t} v(\tau)\, d\tau \]
4.2 Liketsahalo tsa Elektromakenete
Ho electromagnetism, metsoako e kopaneng le yona e thehile mehopolo ya bohlokwa jwalo ka molao wa Gauss le molao wa Ampère. Mohlala, molao wa Gauss bakeng sa tshimo ya motlakase:
\[ \oint_{\partial V} \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{in}}}{\epsilon_0} \]
Ka ho tšoanang, sebakeng sa Hamiltonian bakeng sa litsamaiso tsa thermodynamic, li-integral li sebelisoa ho bala li-microconfigurations tse lumellanang le matla a fanoeng.
5. Moralo oa Boenjiniere
Boenjiniereng, metsoako e kopaneng e sebediswa ho sekaseka kgatello, diphetoho le kabo ya thepa. Mohlala, mekheniking ya thepa, ho bala motsotso wa ho se sebetse hantle ha karolo e tshekaletseng ho hloka motsoako o habeli.
5.1 Motsotso oa Boima
Nako ea ho se sebetse hantle ha sebaka sa A mabapi le mothapo oa y e fanoa ke:
\[ I_y = \int_{A} x^2\, dA \]
Haeba re sekaseka khutlonnetsepa ka bophara \( b \) le bophahamo \( h \), motsotso wa yona wa ho se tsitse ke:
\[ I_y = \int_{0}^{h} \int_{0}^{b} x^2\, dx\, dy = \frac{bh^3}{12} \]
Qetellong, lits'ebeliso tsa li-integral li kholo ebile li akaretsa masimo a mangata. Li-integral li thusa ho rarolla mathata a rarahaneng a kenyeletsang lipalo tse tsoelang pele le liphetoho tse ke keng tsa rarolloa ka ho sebelisa mekhoa e arohaneng. Ka mehlala e kaholimo, re ka bona hore na li-integral li bohlokoa hakae ebile li na le tšusumetso e kae ho sekasekeng le ho rarolleng maemo a fapaneng a bophelo ba sebele. Kutloisiso e felletseng ea li-integral e nolofalletsa bo-rasaense, baenjiniere le litsebi tsa moruo ho theha mehlala, ho sekaseka data, le ho etsa liqeto tse betere.
