Lipotso tsa Mehlala le Puisano ea Thepa ea Li-Definite Integrals
Karolo e tobileng ke mohopolo oa motheo oa lipalo, o thusang haholo lits'ebetsong tse fapaneng tsa lipalo, fisiks le boenjiniere. Sehloohong sena, re tla hlalosa litšobotsi tse ling tsa bohlokoa tsa karolo e tobileng le ho fana ka mehlala le litharollo ho tebisa kutloisiso ea hau ea sehlooho.
Matlotlo a Li-Definite Integrals
Pele re kena mathateng a mohlala, ha re hlahlobeng litšobotsi tse ling tsa motheo tsa likarolo tse tobileng tseo ho leng bohlokoa ho li tseba:
1. Thepa ea Linearity:
– Haeba \( f(x) \) le \( g(x) \) e le mesebetsi e ka kopanngwang mme \( a \) le \( b \) e le di-constant, jwale:
\[
\int_a^b [af(x) + bg(x)] \, dx = a \int_a^bf(x) \, dx + b \int_a^bg(x) \, dx.
\]
2. Karolo e Kopanetsoeng ea Kamehla:
– Haeba \( c \) e le ntho e sa fetoheng, joale:
\[
\int_a^bc \, dx = c(b – a).
\]
3. Litšobotsi tsa ho Eketsa Nako:
\[
\int_a^cf(x) \, dx + \int_c^bf(x) \, dx = \int_a^bf(x) \, dx
\]
4. Phetoho ea Meeli:
\[
\int_a^bf(x) \, dx = – \int_b^af(x) \, dx
\]
5. Lefela Moeling o Tšoanang:
\[
\int_a^af(x) \, dx = 0
\]
Mohlala oa Potso ea 1: Ho Sebelisa Thepa ea Linearity
Mohlala oa mathata:
Bala boleng ba:
\[
\int_0^2 (3x^2 + 2x) \, dx
\]
Puisano:
Sebelisa thepa ea linearity ho arola karolo e kopaneng ka tse peli:
\[
\int_0^2 (3x^2 + 2x) \, dx = \int_0^2 3x^2 \, dx + \int_0^2 2x \, dx
\]
A re bale karolo ea pele ea bohlokoa:
\[
\int_0^2 3x^2 \, dx
\]
\[
= 3 \int_0^2 x^2 \, dx
\]
\[
= 3 \left[ \frac{x^3}{3} \right]_0^2
\]
\[
= 3 \left( \frac{2^3}{3} – \frac{0^3}{3} \right)
\]
\[
= 3 \left( \frac{8}{3} \right)
\]
\[
= 8
\]
Jwale, re bala karolo ya bobedi ya bohlokwa:
\[
\int_0^2 2x \, dx
\]
\[
= 2 \int_0^2 x \, dx
\]
\[
= 2 \left[ \frac{x^2}{2} \right]_0^2
\]
\[
= 2 \ka ho le letshehadi( 1 - 0 \ka ho le letona)
\]
\[
= 2
\]
Kopanya liphetho tse peli:
\[
\int_0^2 (3x^2 + 2x) \, dx = 8 + 2 = 10
\]
Mohlala oa Potso ea 2: Kakaretso ea Constant
Mohlala oa mathata:
Bala boleng ba:
\[
\int_1^4 5 \, dx
\]
Puisano:
Re sebelisa thepa e kopaneng ea li-constants, re ka ngola:
\[
\int_1^4 5 \, dx = 5 \cdot (4 – 1)
\]
\[
= 5 \cdot 3
\]
\[
= 15
\]
Mohlala oa Potso ea 3: Litšobotsi tsa Phetoho ea Moeli
Mohlala oa mathata:
Paka hore:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx
\]
Puisano:
Re qala ka karolo e kopaneng ea \( x^2 \) karolong e pakeng tsa \( [2, 5] \):
\[
\int_2^5 x^2 \, dx = \left[ \frac{x^3}{3} \right]_2^5
\]
\[
= \frac{5^3}{3} – \frac{2^3}{3}
\]
\[
= \frac{125}{3} – \frac{8}{3}
\]
\[
= \frac{117}{3}
\]
\[
= 39
\]
Jwale, ha re baleng karolo ya \( x^2 \) karolong ya karohano \( [5, 2] \) mme re etse bonnete ba hore re fetola letshwao la karabo:
\[
\int_5^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_5^2
\]
\[
= \frac{2^3}{3} – \frac{5^3}{3}
\]
\[
= \frac{8}{3} – \frac{125}{3}
\]
\[
= -\frac{117}{3}
\]
\[
= -39
\]
Ho pakiloe hore:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx.
\]
Mohlala Potso ea 4: Litšobotsi tsa ho Eketsa Nako
Mohlala oa mathata:
Haeba \(\int_2^4 f(x) \, dx = 7\) le \(\int_4^6 f(x) \, dx = 5\) di tsejwa, bala boleng ba \(\int_2^6 f(x) \, dx\).
Puisano:
Ho sebelisa thepa ea ho eketsa nako:
\[
\int_2^6 f(x) \, dx = \int_2^4 f(x) \, dx + \int_4^6 f(x) \, dx
\]
\[
= 7 + 5
\]
\[
= 12
\]
Qetello
Karolo e tobileng e na le litšobotsi tse ngata tsa bohlokoa tse ka re thusang ho rarolla mefuta e fapaneng ea mathata ka katleho e kholoanyane. Sehloohong sena, re buisane ka tse ling tsa litšobotsi tsena tsa motheo 'me ra fana ka mehlala e bontšang kamoo litšobotsi tsena li ka sebelisoang kateng ts'ebetsong. Ka kutloisiso le mokhoa o lekaneng oa ho itloaetsa, u tla khona ho rarolla mathata a tobileng a kopaneng ka kholiseho e kholoanyane.
