Mehlala ea Lipotso le Puisano ea Li-Definite Integrals
Karolo e ikhethileng ke mohopolo oa bohlokoa ho calculus, hangata o sebelisetsoang ho fumana sebaka se ka tlas'a mothinya, ho bala bophahamo ba lintho tse rarahaneng, le bakeng sa lits'ebetso tse ling tse ngata boenjiniere le fisiks. Ho buisana ka karolo e ikhethileng ha ho fane feela ka kutloisiso ea motheo ea mohopolo ona empa hape ho matlafatsa tsebo ea rona ea tlhahlobo ea lipalo. Sengoloa sena se ikemiselitse ho fana ka mehlala ea mathata a ikhethileng a kopaneng hammoho le lipuisano tse qaqileng.
Khopolo ea Motheo ea Kopano e Tiileng
Pele re kena mathateng a mohlala, ha re hlahlobeng mehopolo e meng ea motheo ea li-integral tse tobileng. Se-integral se tobileng, se bontšitsoeng ke \(\int_a^bf(x) \, dx\), se emela sebaka se ka tlas'a sekhutlo sa mosebetsi \(f(x)\) ho tloha ntlheng \(x = a\) ho ea ntlheng \(x = b\).
Ho ya ka dipalo, karolo e ikgethang ho tloha ho \(a\) ho isa ho \(b\) ya mosebetsi \(f(x)\) e ka hlaloswa ka tsela ena:
\[ \int_a^bf(x) \, dx = F(b) – F(a) \]
moo \(F(x)\) e leng antiderivative ya \(f(x)\).
Lipotso tsa Mehlala le Puisano
A re shebeng mehlala e meng ea mathata a bohlokoa le lipuisano tsa 'ona.
Mohlala oa Potso ea 1
Potso:
Bala karolo e tobileng ea mosebetsi \(f(x) = 2x\) ho tloha ho \(x = 1\) ho isa ho \(x = 3\).
Puisano:
Ho rarolla motsoako ona, re qala ka ho fumana antiderivative ea \(f(x) = 2x\).
Sethibela-mafu sa \(2x\) ke:
\[ F(x) = x^2 + C \]
Leha ho le jwalo, di-integral tse tobileng ha re hloke ho kopana ho sa fetoheng \(C\).
Jwale, sebedisa meedi ya di-integral ho bala:
\[ \int_1^3 2x \, dx = F(3) – F(1) \]
Bala boleng ba \(F(x)\) meeding ena:
\[ F(3) = 3^2 = 9 \]
\[ F(1) = 1^2 = 1 \]
Kahoo,
\[ \int_1^3 2x \, dx = 9 – 1 = 8 \]
Mohlala oa Potso ea 2
Potso:
Bala karolo e tobileng ea mosebetsi \(f(x) = x^2 + 1\) ho tloha \(x = 0\) ho isa \(x = 2\).
Puisano:
Fumana antiderivative ea \(f(x) = x^2 + 1\).
Sethibela-mafu sa \(x^2\) ke:
\[ \frac{1}{3}x^3 \]
Ntho e thibelang ho nkuwa ha \(1\) ke \(x\).
Kahoo, antiderivative ea \(f(x)\) ke:
\[ F(x) = \frac{1}{3}x^3 + x \]
Jwale, sebedisa meedi ya di-integral ho bala:
\[ \int_0^2 (x^2 + 1) \, dx = F(2) – F(0) \]
Bala boleng ba \(F(x)\) meeding ena:
\[ F(2) = \frac{1}{3}(2)^3 + 2 = \frac{8}{3} + 2 = \frac{8}{3} + \frac{6}{3} = \frac{14}{3} \]
\[ F(0) = \frac{1}{3}(0)^3 + 0 = 0 \]
Kahoo,
\[ \int_0^2 (x^2 + 1) \, dx = \frac{14}{3} – 0 = \frac{14}{3} \]
Mohlala oa Potso ea 3
Potso:
Bala karolo e tobileng ea mosebetsi \(f(x) = e^x\) ho tloha ho \(x = 1\) ho isa ho \(x = 2\).
Puisano:
Fumana antiderivative ea \(f(x) = e^x\).
Ntho e thibelang ho nkuwa ha \(e^x\) ke \(e^x\).
Jwale, sebedisa meedi ya di-integral ho bala:
\[ \int_1^2 e^x \, dx = F(2) – F(1) \]
Bala boleng ba \(F(x)\) meeding ena:
\[ F(2) = e^2 \]
\[ F(1) = e^1 = e \]
Kahoo,
\[ \int_1^2 e^x \, dx = e^2 – e \]
Mohlala oa Potso ea 4
Potso:
Bala karolo e tobileng ea mosebetsi \(f(x) = \sin(x)\) ho tloha \(x = 0\) ho isa \(x = \pi\).
Puisano:
Fumana antiderivative ea \(f(x) = \sin(x)\).
Ntho e thibelang ho nkuwa ha \(\sin(x)\) ke \(-\cos(x)\).
Jwale, sebedisa meedi ya di-integral ho bala:
\[ \int_0^\pi \sin(x) \, dx = F(\pi) – F(0) \]
Bala boleng ba \(F(x)\) meeding ena:
\[ F(\pi) = -\cos(\pi) = -(-1) = 1 \]
\[ F(0) = -\cos(0) = -1 \]
Kahoo,
\[ \int_0^\pi \sin(x) \, dx = 1 – (-1) = 1 + 1 = 2 \]
Mohlala oa Potso ea 5
Potso:
Bala karolo e tobileng ea mosebetsi \(f(x) = \frac{1}{x}\) ho tloha ho \(x = 1\) ho isa ho \(x = e\).
Puisano:
Fumana antiderivative ea \(f(x) = \frac{1}{x}\).
Ntho e thibelang ho nkuwa ha \(\frac{1}{x}\) ke \(\ln|x|\).
Jwale, sebedisa meedi ya di-integral ho bala:
\[ \int_1^e \frac{1}{x} \, dx = F(e) – F(1) \]
Bala boleng ba \(F(x)\) meeding ena:
\[ F(e) = \ln(e) = 1 \]
\[ F(1) = \ln(1) = 0 \]
Kahoo,
\[ \int_1^e \frac{1}{x} \, dx = 1 – 0 = 1 \]
Qetello
Ka mehlala e kaholimo, re itloaelitse ho fumana likarolo tse tobileng tsa mesebetsi e fapaneng ea motheo. Mohatong o mong le o mong, ho bohlokoa ho qala ka ho fumana antiderivative ebe u sebelisa meeli ea karolo e ka sehloohong ho fumana boleng ba ho qetela.
Dikarolo tse sa fetoheng di bapala karolo ya bohlokwa mafapheng a mangata a thuto le ditshebediso tse sebetsang. Ho utlwisisa mohopolo ona le ho ikwetlisa ka mehlala e fapaneng ho tla matlafatsa bokgoni ba hao ba dipalo haholo.
