Mafunso a Zitsanzo ndi Kukambirana za Makhalidwe a Definite Integrals
Chiganizo chotsimikizika ndi mfundo yofunika kwambiri mu calculus, chothandiza kwambiri pa ntchito zosiyanasiyana mu masamu, fizikisi, ndi uinjiniya. M'nkhaniyi, tifotokoza makhalidwe ena ofunikira a chiganizo chotsimikizika ndikupereka zitsanzo ndi mayankho kuti mumvetse bwino nkhaniyi.
Katundu wa Definite Integrals
Tisanalowe mu zitsanzo za mavuto, tiyeni tiwonenso zina mwa zinthu zofunika kwambiri zomwe ndizofunikira kudziwa:
1. Katundu wa Linearity:
– Ngati \( f(x) \) ndi \( g(x) \) ndi ma function ophatikizana ndipo \( a \) ndi \( b \) ndi ma constants, ndiye kuti:
\[
\int_a^b [af(x) + bg(x)] \, dx = a \int_a^bf(x) \, dx + b \int_a^bg(x) \, dx.
\]
2. Chophatikiza cha Constant:
– Ngati \( c \) ndi chinthu chosasinthika, ndiye kuti:
\[
\int_a^bc \, dx = c(b – a).
\]
3. Kapangidwe ka Kuwonjezera Pakati:
\[
\int_a^cf(x) \, dx + \int_c^bf(x) \, dx = \int_a^bf(x) \, dx
\]
4. Kusintha Malire:
\[
\int_a^bf(x) \, dx = – \int_b^af(x) \, dx
\]
5. Zero Pa Malire Omwewo:
\[
\int_a^af(x) \, dx = 0
\]
Chitsanzo Funso 1: Kugwiritsa Ntchito Katundu wa Linearity
Chitsanzo cha mavuto:
Werengerani mtengo wa:
\[
\int_0^2 (3x^2 + 2x) \, dx
\]
Kukambirana:
Gwiritsani ntchito katundu wa linearity kuti mulekanitse integral m'magawo awiri:
\[
\int_0^2 (3x^2 + 2x) \, dx = \int_0^2 3x^2 \, dx + \int_0^2 2x \, dx
\]
Tiyeni tiwerengere gawo loyamba lofunikira:
\[
\int_0^2 3x^2 \, dx
\]
\[
= 3 \int_0^2 x^2 \, dx
\]
\[
= 3 \kumanzere[ \frac{x^3}{3} \kumanja]_0^2
\]
\[
= 3 \kumanzere( \frac{2^3}{3} – \frac{0^3}{3} \kumanja)
\]
\[
= 3 \kumanzere( \frac{8}{3} \kumanja)
\]
\[
= 8
\]
Tsopano, tikuwerengera gawo lachiwiri:
\[
\int_0^2 2x \, dx
\]
\[
= 2 \int_0^2 x \, dx
\]
\[
= 2 \kumanzere[ \frac{x^2}{2} \kumanja]_0^2
\]
\[
= 2 \kumanzere (1 - 0 \kumanja)
\]
\[
= 2
\]
Phatikizani zotsatira ziwirizi:
\[
\int_0^2 (3x^2 + 2x) \, dx = 8 + 2 = 10
\]
Chitsanzo Funso 2: Kuphatikiza kwa Chokhazikika
Chitsanzo cha mavuto:
Werengerani mtengo wa:
\[
\int_1^4 5 \, dx
\]
Kukambirana:
Pogwiritsa ntchito katundu wofunikira wa zinthu zosasinthika, tikhoza kulemba:
\[
\int_1^4 5 \, dx = 5 \cdot (4 – 1)
\]
\[
= 5 \cdot 3
\]
\[
= 15
\]
Chitsanzo Funso 3: Katundu wa Kusintha kwa Malire
Chitsanzo cha mavuto:
Tsimikizirani kuti:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx
\]
Kukambirana:
Timayamba ndi integral ya \( x^2 \) pa interval \( [2, 5] \):
\[
\int_2^5 x^2 \, dx = \left[ \frac{x^3}{3} \right]_2^5
\]
\[
= \frac{5^3}{3} – \frac{2^3}{3}
\]
\[
= \frac{125}{3} – \frac{8}{3}
\]
\[
= \frac{117}{3}
\]
\[
= 39
\]
Tsopano, tiyeni tiwerengere integral ya \( x^2 \) pa interval \( [5, 2] \) ndikutsimikiza kuti tasintha chizindikiro cha yankho:
\[
\int_5^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_5^2
\]
\[
= \frac{2^3}{3} – \frac{5^3}{3}
\]
\[
= \frac{8}{3} – \frac{125}{3}
\]
\[
= -\frac{117}{3}
\]
\[
= -39
\]
Zatsimikiziridwa kuti:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx.
\]
Chitsanzo Funso 4: Makhalidwe a Kuwonjezera Pakati
Chitsanzo cha mavuto:
Ngati \(\int_2^4 f(x) \, dx = 7\) ndi \(\int_4^6 f(x) \, dx = 5\) zimadziwika, werengerani mtengo wa \(\int_2^6 f(x) \, dx\).
Kukambirana:
Kugwiritsa ntchito kalembedwe ka interval addition:
\[
\int_2^6 f(x) \, dx = \int_2^4 f(x) \, dx + \int_4^6 f(x) \, dx
\]
\[
= 7 + 5
\]
\[
= 12
\]
Mapeto
Chigwirizano chotsimikizika chili ndi makhalidwe ambiri ofunikira omwe angatithandize kuthetsa mavuto osiyanasiyana moyenera. M'nkhaniyi, takambirana zina mwa makhalidwe ofunikirawa ndipo tapereka zitsanzo zomwe zikusonyeza momwe makhalidwewa angagwiritsidwire ntchito. Mukamvetsetsa bwino komanso kuchita zinthu moyenera, mudzatha kuthetsa mavuto otsimikizika molimba mtima.
