Zitsanzo za Mafunso ndi Kukambirana za Definite Integrals
Chiganizo chotsimikizika ndi mfundo yofunika kwambiri mu calculus, yomwe nthawi zambiri imagwiritsidwa ntchito kupeza malo omwe ali pansi pa curve, kuwerengera kuchuluka kwa zinthu zovuta, komanso pazinthu zina zambiri mu uinjiniya ndi fizikisi. Kukambirana za chiganizo chotsimikizika sikuti kumangopereka kumvetsetsa koyambira kwa lingaliro ili komanso kumalimbitsa luso lathu losanthula masamu. Cholinga cha nkhaniyi ndikupereka zitsanzo za mavuto otsimikizika otsimikizika pamodzi ndi zokambirana zatsatanetsatane.
Lingaliro Loyambira la Definite Integral
Tisanalowe mu zitsanzo za mavuto, tiyeni tikambirane mfundo zina zoyambira za ma integral otsimikizika. Ma integral otsimikizika, omwe amasonyezedwa ndi \(\int_a^bf(x) \, dx\), akuyimira dera lomwe lili pansi pa curve ya ntchito \(f(x)\) kuyambira pa point \(x = a\) mpaka pa point \(x = b\).
Mwa masamu, chiganizo chotsimikizika kuyambira \(a\) mpaka \(b\) cha ntchito \(f(x)\) chingafotokozedwe motere:
\[ \int_a^bf(x) \, dx = F(b) – F(a) \]
kumene \(F(x)\) ndiye mankhwala oletsa kufalikira kwa \(f(x)\).
Mafunso ndi Kukambirana Zitsanzo
Tiyeni tiwone zitsanzo zina za mavuto enieni ndi zokambirana zawo.
Chitsanzo cha Funso 1
Funso:
Werengerani chigwirizano chotsimikizika cha ntchito \(f(x) = 2x\) kuyambira \(x = 1\) mpaka \(x = 3\).
Kukambirana:
Kuti tithetse vutoli, choyamba timapeza antiderivative ya \(f(x) = 2x\).
Chotsutsana ndi \(2x\) ndi:
\[ F(x) = x^2 + C \]
Komabe, mu ma integral otsimikizika sitifunikira nthawi zonse ya integration \(C\).
Tsopano, gwiritsani ntchito malire a ma integrals kuti muwerenge:
\[ \int_1^3 2x \, dx = F(3) – F(1) \]
Werengerani mtengo wa \(F(x)\) pa malire awa:
\[ F(3) = 3^2 = 9 \]
\[ F(1) = 1^2 = 1 \]
Choncho,
\[ \int_1^3 2x \, dx = 9 – 1 = 8 \]
Chitsanzo cha Funso 2
Funso:
Werengerani chigwirizano chotsimikizika cha ntchito \(f(x) = x^2 + 1\) kuyambira \(x = 0\) mpaka \(x = 2\).
Kukambirana:
Pezani chotsutsana ndi chochokera ku \(f(x) = x^2 + 1\).
Chotsutsana ndi \(x^2\) ndi:
\[ \frac{1}{3}x^3 \]
Chotsutsa cha \(1\) ndi \(x\).
Kotero, mankhwala oletsa kufalikira kwa \(f(x)\) ndi awa:
\[ F(x) = \frac{1}{3}x^3 + x \]
Tsopano, gwiritsani ntchito malire a ma integrals kuti muwerenge:
\[ \int_0^2 (x^2 + 1) \, dx = F(2) – F(0) \]
Werengerani mtengo wa \(F(x)\) pa malire awa:
\[ F(2) = \frac{1}{3}(2)^3 + 2 = \frac{8}{3} + 2 = \frac{8}{3} + \frac{6}{3} = \frac{14}{3} \]
\[ F(0) = \frac{1}{3}(0)^3 + 0 = 0 \]
Choncho,
\[ \int_0^2 (x^2 + 1) \, dx = \frac{14}{3} – 0 = \frac{14}{3} \]
Chitsanzo cha Funso 3
Funso:
Werengerani chigwirizano chotsimikizika cha ntchito \(f(x) = e^x\) kuyambira \(x = 1\) mpaka \(x = 2\).
Kukambirana:
Pezani chotsutsana ndi chochokera ku \(f(x) = e^x\).
Chotsutsana ndi \(e^x\) ndi \(e^x\).
Tsopano, gwiritsani ntchito malire a ma integrals kuti muwerenge:
\[ \int_1^2 e^x \, dx = F(2) – F(1) \]
Werengerani mtengo wa \(F(x)\) pa malire awa:
\[ F(2) = e^2 \]
\[ F(1) = e^1 = e \]
Choncho,
\[ \int_1^2 e^x \, dx = e^2 – e \]
Chitsanzo cha Funso 4
Funso:
Werengerani chigwirizano chotsimikizika cha ntchito \(f(x) = \sin(x)\) kuchokera \(x = 0\) mpaka \(x = \pi\).
Kukambirana:
Pezani choletsa chochokera ku \(f(x) = \sin(x)\).
Chotsutsana ndi \(\sin(x)\) ndi \(-\cos(x)\).
Tsopano, gwiritsani ntchito malire a ma integrals kuti muwerenge:
\[ \int_0^\pi \sin(x) \, dx = F(\pi) – F(0) \]
Werengerani mtengo wa \(F(x)\) pa malire awa:
\[ F(\pi) = -\cos(\pi) = -(-1) = 1 \]
\[ F(0) = -\cos(0) = -1 \]
Choncho,
\[ \int_0^\pi \sin(x) \, dx = 1 – (-1) = 1 + 1 = 2 \]
Chitsanzo cha Funso 5
Funso:
Werengerani chigwirizano chotsimikizika cha ntchito \(f(x) = \frac{1}{x}\) kuyambira \(x = 1\) mpaka \(x = e\).
Kukambirana:
Pezani chotsutsana ndi \(f(x) = \frac{1}{x}\).
Chotsutsana ndi \(\frac{1}{x}\) ndi \(\ln|x|\).
Tsopano, gwiritsani ntchito malire a ma integrals kuti muwerenge:
\[ \int_1^e \frac{1}{x} \, dx = F(e) – F(1) \]
Werengerani mtengo wa \(F(x)\) pa malire awa:
\[ F(e) = \ln(e) = 1 \]
\[ F(1) = \ln(1) = 0 \]
Choncho,
\[ \int_1^e \frac{1}{x} \, dx = 1 – 0 = 1 \]
Mapeto
Kudzera mu zitsanzo zomwe zili pamwambapa, tachita kafukufuku wopeza zinthu zofunika kwambiri pa ntchito zosiyanasiyana zoyambira. Pa sitepe iliyonse, ndikofunikira kupeza kaye chinthu chotsutsana ndi chinthucho kenako kugwiritsa ntchito malire a chinthucho kuti tipeze phindu lomaliza.
Ma integral okhazikika amachita gawo lofunikira kwambiri m'magawo ambiri ophunzirira ndi ntchito zothandiza. Kumvetsetsa lingaliro ili ndikuchita ndi zitsanzo zosiyanasiyana kudzalimbitsa kwambiri luso lanu la masamu.
