Mienzaniso yeMibvunzo neKukurukurirana kweDefinite Integrals
Chinhu chinobatanidzwa (definite integral) ipfungwa huru mukuverenga, inowanzoshandiswa kuwana nzvimbo iri pasi pemugero, kuverenga huwandu hwezvinhu zvakaoma, uye kune mamwe mashandisirwo akawanda muinjiniya nefizikisi. Kukurukura nezvechinhu chinobatanidzwa (definite integral) hakungopi chete kunzwisisa kwekutanga kwepfungwa iyi asiwo kunosimbisa hunyanzvi hwedu hwekuongorora masvomhu. Chinyorwa chino chine chinangwa chekupa mienzaniso yezvinetso zvinowirirana pamwe chete nekukurukurirana kwakadzama.
Pfungwa Yekutanga yeDefinite Integral
Tisati tapinda mumatambudziko emuenzaniso, ngationgororei mamwe mazano ekutanga ezviverengero zvinotsanangurwa. Chiverengero chinonzwisisika, chinonongedzwa ne \(\int_a^bf(x) \, dx\), chinomiririra nzvimbo iri pasi pekongiri yebasa \(f(x)\) kubva papoindi \(x = a\) kusvika papoindi \(x = b\).
Pamasvomhu, chiverengero chakajeka kubva pa \(a\) kusvika pa \(b\) chebasa \(f(x)\) chinogona kuratidzwa seizvi:
\[ \int_a^bf(x) \, dx = F(b) – F(a) \]
apo \(F(x)\) iri mushonga unodzivirira kubva pa \(f(x)\).
Mibvunzo yemuenzaniso nekukurukurirana
Ngatitarisei mimwe mienzaniso yezvinetso zvakakosha uye hurukuro dzazvo.
Muenzaniso Mubvunzo 1
Mubvunzo:
Verenga huwandu hwebasa \(f(x) = 2x\) kubva \(x = 1\) kusvika \(x = 3\).
Kukurukurirana:
Kuti tigadzirise chinhu ichi, tinotanga tawana chinhu chinopesana ne \(f(x) = 2x\).
Chinhu chinopesana ne \(2x\) ndechekuti:
\[ F(x) = x^2 + C \]
Zvisinei, muzvikamu zvakati wandei hazvidi kuti pave nekubatanidzwa kwacho (constant of integration) (C).
Zvino, shandisa miganhu yezvikamu zvakakosha kuverenga:
\[ \int_1^3 2x \, dx = F(3) – F(1) \]
Verenga kukosha kwe \(F(x)\) pamiganhu iyi:
\[ F(3) = 3^2 = 9 \]
\[ F(1) = 1^2 = 1 \]
Saka,
\[ \int_1^3 2x \, dx = 9 – 1 = 8 \]
Muenzaniso Mubvunzo 2
Mubvunzo:
Verenga huwandu hwebasa \(f(x) = x^2 + 1\) kubva \(x = 0\) kusvika \(x = 2\).
Kukurukurirana:
Tsvaga chinhu chinopesana ne \(f(x) = x^2 + 1\).
Chinhu chinopesana ne \(x^2\) ndechekuti:
\[ \frac{1}{3}x^3 \]
Chinhu chinopesana ne \(1\) ndi \(x\).
Saka, mushonga unodzivirira kubva pa \(f(x)\) ndewekuti:
\[ F(x) = \frac{1}{3}x^3 + x \]
Zvino, shandisa miganhu yezvikamu zvakakosha kuverenga:
\[ \int_0^2 (x^2 + 1) \, dx = F(2) – F(0) \]
Verenga kukosha kwe \(F(x)\) pamiganhu iyi:
\[ F(2) = \frac{1}{3}(2)^3 + 2 = \frac{8}{3} + 2 = \frac{8}{3} + \frac{6}{3} = \frac{14}{3} \]
\[ F(0) = \frac{1}{3}(0)^3 + 0 = 0 \]
Saka,
\[ \int_0^2 (x^2 + 1) \, dx = \frac{14}{3} – 0 = \frac{14}{3} \]
Muenzaniso Mubvunzo 3
Mubvunzo:
Verenga huwandu hwebasa \(f(x) = e^x\) kubva \(x = 1\) kusvika \(x = 2\).
Kukurukurirana:
Tsvaga chinhu chinopesana ne \(f(x) = e^x\).
Chinhu chinopesana ne \(e^x\) ndi \(e^x\).
Zvino, shandisa miganhu yezvikamu zvakakosha kuverenga:
\[ \int_1^2 e^x \, dx = F(2) – F(1) \]
Verenga kukosha kwe \(F(x)\) pamiganhu iyi:
\[ F(2) = e^2 \]
\[ F(1) = e^1 = e \]
Saka,
\[ \int_1^2 e^x \, dx = e^2 – e \]
Muenzaniso Mubvunzo 4
Mubvunzo:
Verenga huwandu hwebasa \(f(x) = \sin(x)\) kubva \(x = 0\) kusvika \(x = \pi\).
Kukurukurirana:
Tsvaga chinhu chinodzivirira kubva pa \(f(x) = \sin(x)\).
Chinhu chinopesana ne \(\sin(x)\) ndi \(-\cos(x)\).
Zvino, shandisa miganhu yezvikamu zvakakosha kuverenga:
\[ \int_0^\pi \sin(x) \, dx = F(\pi) – F(0) \]
Verenga kukosha kwe \(F(x)\) pamiganhu iyi:
\[ F(\pi) = -\cos(\pi) = -(-1) = 1 \]
\[ F(0) = -\cos(0) = -1 \]
Saka,
\[ \int_0^\pi \sin(x) \, dx = 1 – (-1) = 1 + 1 = 2 \]
Muenzaniso Mubvunzo 5
Mubvunzo:
Verenga huwandu hwebasa \(f(x) = \frac{1}{x}\) kubva \(x = 1\) kusvika \(x = e\).
Kukurukurirana:
Tsvaga chinhu chinopesana ne \(f(x) = \frac{1}{x}\).
Chinhu chinopesana ne \(\frac{1}{x}\) ndi \(\ln|x|\).
Zvino, shandisa miganhu yezvikamu zvakakosha kuverenga:
\[ \int_1^e \frac{1}{x} \, dx = F(e) – F(1) \]
Verenga kukosha kwe \(F(x)\) pamiganhu iyi:
\[ F(e) = \ln(e) = 1 \]
\[ F(1) = \ln(1) = 0 \]
Saka,
\[ \int_1^e \frac{1}{x} \, dx = 1 – 0 = 1 \]
Mhedziso
Kuburikidza nemienzaniso iri pamusoro apa, takadzidzira kutsvaga zvinhu zvakakosha zvemabasa akasiyana-siyana ekutanga. Mudanho rega rega, zvakakosha kutanga wawana chinhu chinodzivirira kubva pachinhu ichocho wobva washandisa miganhu yechinhu ichocho kuti uwane kukosha kwekupedzisira.
Zvinhu zvinosanganisa zvinhu zvine basa guru muzvidzidzo zvakawanda uye mashandisirwo azvo. Kunzwisisa pfungwa iyi uye kudzidzira nemienzaniso yakasiyana-siyana kuchasimbisa hunyanzvi hwako hwemasvomhu zvakanyanya.
