Mibvunzo yeMienzaniso neKukurukurirana kweZvinhu zveDefinite Integrals
Chinongedzo chepfungwa chinobatsira zvikuru mukuverenga, chinobatsira zvikuru mukushandiswa kwakasiyana-siyana mumasvomhu, fizikisi, uye mainjiniya. Muchinyorwa chino, tichatsanangura zvimwe zvinhu zvakakosha zvechinongedzo chepfungwa chinobatanidza uye tokupa mienzaniso nemhinduro kuti tiwedzere kunzwisisa kwako musoro wenyaya.
Zvivakwa zveDefinite Integrals
Tisati tapinda mumatambudziko emuenzaniso, ngationgororei zvimwe zvinhu zvepakutanga zvema integrals akakosha kuziva:
1. Hunhu hweLinearity:
– Kana \( f(x) \) uye \( g(x) \) ari mabasa anobatanidzwa uye \( a \) uye \( b \) ari ma constants, saka:
\[
\int_a^b [af(x) + bg(x)] \, dx = a \int_a^bf(x) \, dx + b \int_a^bg(x) \, dx.
\]
2. Chinhu Chinokosha cheConstant:
– Kana \( c \) iri chinhu chisingachinji, saka:
\[
\int_a^bc \, dx = c(b – a).
\]
3. Hunhu hwekuwedzera kwechikamu:
\[
\int_a^cf(x) \, dx + \int_c^bf(x) \, dx = \int_a^bf(x) \, dx
\]
4. Kuchinja Miganhu:
\[
\int_a^bf(x) \, dx = – \int_b^af(x) \, dx
\]
5. Zero Pamuganhu Wakafanana:
\[
\int_a^af(x) \, dx = 0
\]
Muenzaniso Mubvunzo 1: Kushandisa Linearity Property
Muenzaniso wematambudziko:
Verenga kukosha kwe:
\[
\int_0^2 (3x^2 + 2x) \, dx
\]
Kukurukurirana:
Shandisa chivakwa che linearity kupatsanura integral kuita zviviri:
\[
\int_0^2 (3x^2 + 2x) \, dx = \int_0^2 3x^2 \, dx + \int_0^2 2x \, dx
\]
Ngativerengei chinhu chekutanga chakakosha:
\[
\int_0^2 3x^2 \, dx
\]
\[
= 3 \int_0^2 x^2 \, dx
\]
\[
= 3 \kuruboshwe[ \frac{x^3}{3} \kurudyi]_0^2
\]
\[
= 3 \kuruboshwe( \frac{2^3}{3} – \frac{0^3}{3} \kurudyi)
\]
\[
= 3 \kuruboshwe( \frac{8}{3} \kurudyi)
\]
\[
= 8
\]
Zvino, tinoverenga chikamu chechipiri:
\[
\int_0^2 2x \, dx
\]
\[
= 2 \int_0^2 x \, dx
\]
\[
= 2 \kuruboshwe[ \frac{x^2}{2} \kurudyi]_0^2
\]
\[
= 2 \kuruboshwe (1 - 0 \kurudyi)
\]
\[
= 2
\]
Sanganisa mhedzisiro mbiri idzi:
\[
\int_0^2 (3x^2 + 2x) \, dx = 8 + 2 = 10
\]
Muenzaniso Mubvunzo 2: Kubatanidzwa kweConstant
Muenzaniso wematambudziko:
Verenga kukosha kwe:
\[
\int_1^4 5 \, dx
\]
Kukurukurirana:
Tichishandisa pfuma inobatanidza ye constants, tinogona kunyora:
\[
\int_1^4 5 \, dx = 5 \cdot (4 – 1)
\]
\[
= 5 \cdot 3
\]
\[
= 15
\]
Muenzaniso Mubvunzo 3: Hunhu hweKuchinja Kwemuganhu
Muenzaniso wematambudziko:
Ratidza kuti:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx
\]
Kukurukurirana:
Tinotanga nekubatanidzwa kwe \( x^2 \) pane chikamu \( [2, 5] \):
\[
\int_2^5 x^2 \, dx = \kuruboshwe[ \frac{x^3}{3} \kurudyi]_2^5
\]
\[
= \frac{5^3}{3} – \frac{2^3}{3}
\]
\[
= \frac{125}{3} – \frac{8}{3}
\]
\[
= \frac{117}{3}
\]
\[
= 39
\]
Zvino, ngativerengei kukosha kwe \( x^2 \) pane interval \( [5, 2] \) uye tive nechokwadi chekudzosera chiratidzo chemhinduro kumashure:
\[
\int_5^2 x^2 \, dx = \kuruboshwe[ \frac{x^3}{3} \kurudyi]_5^2
\]
\[
= \frac{2^3}{3} – \frac{5^3}{3}
\]
\[
= \frac{8}{3} – \frac{125}{3}
\]
\[
= -\frac{117}{3}
\]
\[
= -39
\]
Zvinoratidzwa kuti:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx.
\]
Muenzaniso Mubvunzo 4: Hunhu hwekuwedzera kwepakati
Muenzaniso wematambudziko:
Kana \(\int_2^4 f(x) \, dx = 7\) uye \(\int_4^6 f(x) \, dx = 5\) zvichizivikanwa, verenga kukosha kwe \(\int_2^6 f(x) \, dx\).
Kukurukurirana:
Kushandisa pfuma yekuwedzera nguva:
\[
\int_2^6 f(x) \, dx = \int_2^4 f(x) \, dx + \int_4^6 f(x) \, dx
\]
\[
= 7 + 5
\]
\[
= 12
\]
Mhedziso
Chinhu chinobatanidza chine hunhu hwakawanda hunokosha hunogona kutibatsira kugadzirisa matambudziko akasiyana-siyana zvinobudirira. Muchinyorwa chino, takurukura zvimwe zvehunhu uhwu uye tapa mienzaniso inoratidza kuti hunhu uhwu hunogona kushandiswa sei mukuita. Nekunzwisisa kwakakwana uye kudzidzira, uchakwanisa kugadzirisa matambudziko akabatana ane chivimbo chikuru.
