Imibuzo Yezibonelo kanye Nengxoxo Yezakhiwo Ze-Definite Integrals
I-integral eqondile ingumqondo oyisisekelo ekubaleni, iwusizo kakhulu ezinhlobonhlobo zezicelo kuzibalo, ifiziksi, kanye nobunjiniyela. Kulesi sihloko, sizochaza ezinye zezakhiwo ezibalulekile ze-integral eqondile futhi sinikeze izibonelo nezixazululo zokujulisa ukuqonda kwakho ngesihloko.
Izakhiwo ze-Definite Integrals
Ngaphambi kokuthi singene ezinkingeni zesibonelo, ake sibukeze ezinye zezakhiwo eziyisisekelo zama-integral aqondile okubalulekile ukuzazi:
1. Impahla Yokulingana:
– Uma \( f(x) \) kanye \( g(x) \) kuyimisebenzi ehlanganisiwe kanye \( a \) kanye \( b \) kuyizinto ezihlala njalo, khona-ke:
\[
\int_a^b [af(x) + bg(x)] \, dx = a \int_a^bf(x) \, dx + b \int_a^bg(x) \, dx.
\]
2. Okuhlanganisiwe kwe-Constant:
– Uma i-\( c \) iyinto engaguquki, khona-ke:
\[
\int_a^bc \, dx = c(b – a).
\]
3. Izakhiwo Zokwengeza Isikhawu:
\[
\int_a^cf(x) \, dx + \int_c^bf(x) \, dx = \int_a^bf(x) \, dx
\]
4. Ukuguqulwa Kwemingcele:
\[
\int_a^bf(x) \, dx = – \int_b^af(x) \, dx
\]
5. U-Zero Emkhawulweni Ofanayo:
\[
\int_a^af(x) \, dx = 0
\]
Isibonelo Umbuzo 1: Ukusebenzisa Impahla Yomugqa
Isibonelo sezinkinga:
Bala inani le:
\[
\int_0^2 (3x^2 + 2x) \, dx
\]
Ingxoxo:
Sebenzisa isici se-linearity ukuhlukanisa i-integral ibe yizimbili:
\[
\int_0^2 (3x^2 + 2x) \, dx = \int_0^2 3x^2 \, dx + \int_0^2 2x \, dx
\]
Ake sibale i-integral yokuqala:
\[
\int_0^2 3x^2 \, dx
\]
\[
= 3 \int_0^2 x^2 \, dx
\]
\[
= 3 \kwesobunxele[ \frac{x^3}{3} \kwesokudla]_0^2
\]
\[
= 3 \kwesobunxele( \frac{2^3}{3} – \frac{0^3}{3} \kwesokudla)
\]
\[
= 3 \kwesobunxele( \frac{8}{3} \kwesokudla)
\]
\[
= 8
\]
Manje, sibala i-integral yesibili:
\[
\int_0^2 2x \, dx
\]
\[
= 2 \int_0^2 x \, dx
\]
\[
= 2 \kwesobunxele[ \frac{x^2}{2} \kwesokudla]_0^2
\]
\[
= 2 \kwesobunxele( 1 – 0 \kwesokudla)
\]
\[
= 2
\]
Hlanganisa imiphumela emibili:
\[
\int_0^2 (3x^2 + 2x) \, dx = 8 + 2 = 10
\]
Isibonelo Umbuzo 2: Okuhlanganisiwe kwe-Constant
Isibonelo sezinkinga:
Bala inani le:
\[
\int_1^4 5 \, dx
\]
Ingxoxo:
Sisebenzisa impahla ehlanganisiwe yama-constants, singabhala:
\[
\int_1^4 5 \, dx = 5 \cdot (4 – 1)
\]
\[
= 5 \cdot 3
\]
\[
= 15
\]
Isibonelo Umbuzo 3: Izakhiwo Zokushintsha Komkhawulo
Isibonelo sezinkinga:
Fakazela ukuthi:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx
\]
Ingxoxo:
Siqala ngokuhlanganiswa kwe- \( x^2 \) esikhaleni \( [2, 5] \):
\[
\int_2^5 x^2 \, dx = \kwesobunxele[ \frac{x^3}{3} \kwesokudla]_2^5
\]
\[
= \frac{5^3}{3} – \frac{2^3}{3}
\]
\[
= \frac{125}{3} – \frac{8}{3}
\]
\[
= \frac{117}{3}
\]
\[
= 39
\]
Manje, ake sibale i-integral ye-\( x^2 \) ku-interval \( [5, 2] \) bese siqinisekisa ukuthi sibuyisela emuva uphawu lwempendulo:
\[
\int_5^2 x^2 \, dx = \kwesobunxele[ \frac{x^3}{3} \kwesokudla]_5^2
\]
\[
= \frac{2^3}{3} – \frac{5^3}{3}
\]
\[
= \frac{8}{3} – \frac{125}{3}
\]
\[
= -\frac{117}{3}
\]
\[
= -39
\]
Kufakazelwe ukuthi:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx.
\]
Isibonelo Umbuzo 4: Izakhiwo Zokwengeza Kwesikhawu
Isibonelo sezinkinga:
Uma kuziwa ukuthi \(\int_2^4 f(x) \, dx = 7\) kanye \(\int_4^6 f(x) \, dx = 5\), bala inani \(\int_2^6 f(x) \, dx\).
Ingxoxo:
Ukusebenzisa isici sokwengeza isikhawu:
\[
\int_2^6 f(x) \, dx = \int_2^4 f(x) \, dx + \int_4^6 f(x) \, dx
\]
\[
= 7 + 5
\]
\[
= 12
\]
Isiphetho
I-integral eqondile inezimpawu eziningi ezibalulekile ezingasisiza ukuxazulula izinhlobo ezahlukene zezinkinga ngendlela ephumelela kakhudlwana. Kulesi sihloko, sixoxe ngezinye zalezi zimpawu eziyisisekelo futhi sanikeza izibonelo ezibonisa ukuthi lezi zimpawu zingasetshenziswa kanjani ekusebenzeni. Ngokuqonda okwanele kanye nokuzijwayeza, uzokwazi ukuxazulula izinkinga eziqondile eziqondile ngokuzethemba okukhulu.
