Mehlala ea Lipotso le Puisano ea Ts'ebeliso ea Li-Integrals ho Bala Sebaka sa Sefofane se Sephara
Thutong ea lipalo, li-integral hangata li fumanoa ho calculus. E 'ngoe ea lits'ebetso tse tsebahalang haholo tsa li-integral ke ho bala sebaka tlas'a kobeho kapa sefofane. Sengoloa sena se tla tšohla mathata a 'maloa a mehlala le ho tšohla ts'ebeliso ea li-integral ho bala sebaka sa sefofane.
Selelekela ho Khopolo-taba
Pele re fetela bothateng ba mohlala, ha re hlahlobeng mohopolo oa motheo oa ho bala sebaka se ka tlas'a mothapo o sebelisang li-integral. Haeba re na le ts'ebetso f(x) e tsoelang pele karolong [a, b], joale sebaka se ka tlas'a mothapo y = f(x) ho tloha x = a ho isa x = b ke:
\[ L = \int_{a}^{b} f(x) \, dx \]
Ho ya ka jeometri, sena se bolela hore re akaretsa sebaka sa khutlonnetsepa e tshesane haholo ho tloha ho x = a ho isa ho x = b.
Mohlala oa Potso ea 1
Soal
Bala sebaka se ka tlas'a mothapo y = x² karolong e pakeng tsa [1, 3].
Puisano
Ho bala sebaka, re sebelisa motsoako o kopaneng:
\[ L = \int_{1}^{3} x^2 \, dx \]
Re qala ka ho fumana antiderivative ea \( x^2 \). Antiderivative ea \( x^2 \) ke \( \frac{x^3}{3} \). Ebe karolo e kopaneng e ba:
\[ L = \left[ \frac{x^3}{3} \right]_{1}^{3} \]
Hopola hore re tlameha ho lekola antiderivative ho latela meeli ea integral:
\[ L = \left( \frac{3^3}{3} \right) – \left( \frac{1^3}{3} \right) \]
\[ L = \left( \frac{27}{3} \right) – \left( \frac{1}{3} \right) \]
\[ L = 9 – \frac{1}{3} \]
\[ L = \frac{27}{3} – \frac{1}{3} \]
\[ L = \frac{26}{3} \]
Kahoo, sebaka se ka tlas'a mothapo y = x² ho tloha ho x = 1 ho isa ho x = 3 ke:
\[ \frac{26}{3} \, \yuniti ya sebaka sa mongolo} \]
Mohlala oa Potso ea 2
Soal
Fumana sebaka sa sebaka se moeditsweng ke mothapo y = x³ le mela x = 1 le x = 2.
Puisano
Ho bala sebaka, re sebelisa motsoako o kopaneng:
\[ L = \int_{1}^{2} x^3 \, dx \]
Jwalo ka tlwaelo, re qala ka ho fumana antiderivative ya \( x^3 \). Antiderivative ya \( x^3 \) ke \( \frac{x^4}{4} \). Karolo e kopaneng e ba:
\[ L = \left[ \frac{x^4}{4} \right]_{1}^{2} \]
Lekola meeli ea karolo e kopaneng:
\[ L = \left( \frac{2^4}{4} \right) – \left( \frac{1^4}{4} \right) \]
\[ L = \left( \frac{16}{4} \right) – \left( \frac{1}{4} \right) \]
\[ L = 4 – \frac{1}{4} \]
\[ L = \frac{16}{4} – \frac{1}{4} \]
\[ L = \frac{15}{4} \]
Kahoo, sebaka se ka tlas'a mothapo y = x³ ho tloha x = 1 ho isa ho x = 2 ke:
\[ \frac{15}{4} \, \yuniti ya sebaka sa mongolo} \]
Mohlala oa Potso ea 3
Soal
Fumana sebaka sa sebaka se moeditsweng ke di-curve y = x² + 1 le y = 2x + 2 karolong ya x = 0 ho isa ho x = 1.
Puisano
Taba ea pele, re hloka ho fumana lintlha tsa khokahano ho fumana meeli ea kopanyo. Tharollo ea \( x^2 + 1 = 2x + 2 \):
\[ x^2 + 1 = 2x + 2 \]
\[ x^2 – 2x – 1 = 0 \]
Ho sebelisa foromo ea quadratic:
\[ x = \frac{2 \pm \sqrt{4 + 4}}{2} \]
\[ x = \frac{2 \pm \sqrt{8}}{2} \]
\[ x = \frac{2 \pm 2\sqrt{2}}{2} \]
\[ x = 1 \pm \sqrt{2} \]
Leha ho le jwalo, bakeng sa meedi e ka hodimo le e ka tlase pakeng tsa 0 le 1, ha ho hlokahale hore re sebedise tharollo ya quadratic, re hloka feela moedi o tlwaelehileng wa integral ho tloha ho 0 ho isa ho 1. Ka mora moo, bala sebaka sa curve e ka hodimo ya y ho tlosa curve e ka tlase ya y ho latela meedi ena:
\[ L = \int_{0}^{1} [(2x + 2) – (x^2 + 1)] \, dx \]
Ho nolofatsa mosebetsi:
\[ L = \int_{0}^{1} (2x + 2 – x^2 – 1) \, dx \]
\[ L = \int_{0}^{1} (-x^2 + 2x + 1) \, dx \]
Ka mor'a moo, re fumana antiderivative:
Sethibela-mafu sa \( (-x^2) \) ke \( -\frac{x^3}{3} \),
Sethibela-mafu sa \( (2x) \) ke \( x^2 \),
Ntho e thibelang ho nkuwa ha \( (1) \) ke \( x \).
E le hore,
\[ L = \left. \left(-\frac{x^3}{3} + x^2 + x \right) \right|_0^1 \]
Tlhahlobo e latelang:
\[ L = \left[ -\frac{1^3}{3} + 1^2 + 1 \right] – \left[ -\frac{0^3}{3} + 0^2 + 0 \right] \]
\[ L = \left[ -\frac{1}{3} + 1 + 1 \right] – \left[ 0 \right] \]
\[ L = -\frac{1}{3} + 2 \]
\[ L = \frac{6}{3} – \frac{1}{3} \]
\[ L = \frac{5}{3} \]
Kahoo, sebaka sa sebaka se moeditsweng ke di-curve y = x² + 1 le y = 2x + 2 karolong [0, 1] ke:
\[ \frac{5}{3} \, \yuniti ya sebaka sa mongolo} \]
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Ho tsoa mehlaleng e kaholimo, re ka bona kamoo li-integral li ka sebelisoang ho bala sebaka se ka tlas'a kotopo kapa pakeng tsa li-curve tse peli. Ka kutloisiso e nepahetseng ea likhopolo tsa motheo tsa li-integral le mekhoa ea antiderivative, ho bala libaka tsena ho ba le mokhoa o hlophisehileng haholo le o sebetsang hantle. Re tšepa hore sengoloa sena se ekelitse kutloisiso ea rona ea ts'ebeliso ea li-integral lefatšeng la 'nete, haholo-holo tšimong ea ho lekanya sebaka sa libaka tsa sefofane.
