Tlhaloso e Ikhethang ka ho Feletseng: Tlhaloso, Khopolo-taba le Tšebeliso
Integral ke e 'ngoe ea likhopolo tsa motheo ka har'a calculus e bapalang karolo ea bohlokoa haholo mafapheng a fapaneng a saense, ho kenyeletsoa lipalo, fisiks, boenjiniere le moruo. Integral e tobileng ke mofuta oa integral o nang le meeli e itseng ea kopanyo, e leng moeli o tlase le o ka holimo, o tšoaeang nako ea kopanyo. Ho fapana le integral e sa lekanyetsoang e hlahisang mesebetsi ea antiderivative, integral e tobileng e na le boleng ba lipalo 'me hangata e sebelisoa ho bala sebaka tlas'a kobeho, bophahamo ba lintho tse tiileng tsa phetoho, le lits'ebetso tse ling tse fapaneng tse sebetsang.
Tlhaloso ea Definite Integral
Karolo e tobileng ea mosebetsi \( f(x) \) karolong ea karohano \([a, b]\) e hlalosoa e le:
\[ \int_{a}^{b} f(x) \, dx \]
Mona, \( a \) le \( b \) ke meedi e tlase le e ka hodimo ya kopanyo, ka ho latellana. Kopanyo ena e hlahisa palo e emelang pokello ya boleng ba mosebetsi \( f(x) \) ho mefuta \( a \) ho isa ho \( b \). Ho ya ka jeometri, motswako o tobileng o ka hlaloswa e le sebaka se moeditsweng ke mothinya \( y = f(x) \), mothapo wa x, le mela e otlolohileng \( x = a \) le \( x = b \).
Khopolo ea Motheo ea Kopano e Tiileng
Likhopolo-taba tsa Motheo tsa Calculus
Khopolo-taba ea Motheo ea Calculus e hokahanya khopolo-taba ea lintho tse kopaneng le khopolo-taba ea lintho tse tsoang ho tsona (phapang). Khopolo-taba ena e arotsoe likarolo tse peli:
1. Karolo ea Pele ea Theorem: Haeba \( F \) e le mosebetsi o khahlanong le ho ntšoa ha oona (mosebetsi oa pele) oa mosebetsi \( f \) karolong ea nako \([a, b]\), joale:
\[ \int_{a}^{b} f(x) \, dx = F(b) – F(a) \]
Karolo ena e bonts'a hore karolo e ikhethileng e ka baloa ka ho fumana antiderivative ea \( f(x) \), ebe ho baloa phapang lipakeng tsa boleng ba antiderivative meeling e holimo le e tlase.
2. Karolo ea Bobeli ea Theorem: Haeba \( f \) e le mosebetsi o tsoelang pele ho \([a, b]\) le \( F(x) \) e le mosebetsi o hlalosoang e le:
\[ F(x) = \int_{a}^{x} f(t) \, dt \]
ebe \( F'(x) = f(x) \). Sena se bontsha hore derivative ya karolo e kopaneng ya mosebetsi e lekana le mosebetsi ka bowona.
Mokhoa oa ho Bala
Palo ea tlhahlobo ea likarolo tse tobileng hangata e kenyelletsa mehato e 'meli e meholo:
– Fumana antiderivative \( F(x) \) ya mosebetsi o fanweng \( f(x) \).
– Bala boleng ba \( F \) meeling e ka holimo le e ka tlase ea kopanyo, ebe u fumana phapang ho fumana sephetho sa kopanyo.
Mohlala, a re re re batla ho bala \( \int_{2}^{5} 3x^2 \, dx \).
1. Sethibela-mafu sa \( 3x^2 \) ke \( F(x) = x^3 \).
2. Bala \( F \) meeling e ka holimo le e ka tlase:
\[ F(5) = 5^3 = 125 \]
\[ F(2) = 2^3 = 8 \]
Kahoo, \[ \int_{2}^{5} 3x^2 \, dx = 125 – 8 = 117 \]
Likopo tse Ikhethang tse Kopaneng
Sebaka se ka Tlas'a Mokokotlo
E 'ngoe ea lits'ebetso tse tloaelehileng tsa motsoako o hlakileng ke ho bala sebaka se tlas'a mothinya. A re re re batla ho bala sebaka se tlas'a mothinya \( y = f(x) \) ho tloha \( x = a \) ho ea ho \( x = b \). Re ka sebelisa motsoako o hlakileng ho fumana sebaka sena:
\[ \mongolo{Sebaka} = \int_{a}^{b} f(x) \, dx \]
Bophahamo ba Lintho tse Potolohang
Metswako e tobileng e ka boela ea sebelisoa ho bala bophahamo ba lintho tse hlahang ka lebaka la ho potoloha ha sekhutlo se potolohileng x-axis kapa y-axis. Mekhoa e sebelisoang hangata ke mokhoa oa disc le mokhoa oa cylinder-shell.
Mokhoa oa Disc
A re re re na le mothapo \( y = f(x) \) mme re batla ho potoloha mothapo ona ho potoloha x-axis ho tloha \( x = a \) ho ya ho \( x = b \). Bophahamo ba ntho e hlahang bo ka balwa ho sebediswa mothapo o itseng ka tsela e latelang:
\[ V = \pi \int_{a}^{b} [f(x)]^2 \, dx \]
Mokhoa oa Letlalo la Tube
Haeba re batla ho potoloha mothapo \( x = g(y) \) ho potoloha mothapo wa y ho tloha \( y = c \) ho ya ho \( y = d \), bophahamo ba yona bo ka balwa ho sebediswa:
\[ V = 2\pi \int_{c}^{d} y \, g(y) \, dy \]
Lisebelisoa tse ling
Fisiks, di-integral tse tobileng hangata di sebediswa ho bala bongata bo fapaneng jwalo ka mosebetsi o entsweng ke matla \( F(x) \) hole \( x \), o hlaloswang ka tsela ena:
\[ W = \int_{a}^{b} F(x) \, dx \]
Moruong, likarolo tse kopaneng li ka sebelisoa ho bala kakaretso ea lekeno kapa litšenyehelo ka nako e itseng, ho latela mosebetsi oa lekeno kapa litšenyehelo ka yuniti ea nako.
Boleng ba Lipalo: Mokhoa oa ho Hakanya
Ha mosebetsi \( f(x) \) o rarahane kapa o se na antiderivative e nepahetseng, mekgwa ya dipalo e sebediswa ho bala integral. Mekhoa e tlwaelehileng e sebediswang hangata e kenyeletsa:
– Mokhoa oa Riemann: E hakanya karolo e kopaneng ka ho akaretsa libaka tsa likhutlonnetsepa tse ka tlas'a mothinya.
– Mokhoa oa Trapezoidal: E hakanya karolo e kopaneng ka ho eketsa libaka tsa trapezoidal tse ka tlas'a mothinya.
– Mokhoa oa Simpson: O sebelisa polynomial ea quadratic ho hakanya sebaka se ka tlas'a mothinya.
Mohlala, mokhoa oa trapezoidal oa ho bala \( \int_{a}^{b} f(x) \, dx \) ka likarohano \( n \) ke:
\[ \int_{a}^{b} f(x) \, dx \approx \frac{ba}{2n} \left[f(x_0) + 2 \sum_{k=1}^{n-1} f(x_k) + f(x_n)\right] \]
moo \( x_0, x_1, …, x_n \) e leng dintlha tse arolang tsa karohano \([a, b]\).
Qetello
Karolo e ikhethileng ke mohopolo oa motheo oa lipalo tse nang le lits'ebetso tse pharaletseng masimong a fapaneng. Ho tloha ho baleng sebaka se ka tlas'a mothapo ho isa ho bophahamo ba lintho tse tiileng tsa phetoho le ho sekaseka bongata ba 'mele le ba moruo, karolo e ikhethileng ke sesebelisoa se matla lipalo tse fapaneng. Re sebelisa mekhoa ea tlhahlobo le ea lipalo, re ka lekola likarolo tse ikhethileng ho fumana liphetho tse nepahetseng le tse sebetsang maemong a sebele. Kutloisiso e felletseng ea likarolo tse ikhethileng e bula monyako oa ho rarolla mathata a mangata a rarahaneng a amanang le mesebetsi le libaka.
