Izibonelo Zemibuzo Nengxoxo Yama-Definite Integrals
I-integral eqondile ingumqondo oyinhloko ekubaleni, evame ukusetshenziselwa ukuthola indawo ngaphansi kwejika, ukubala ivolumu yezinto eziyinkimbinkimbi, kanye nezinye izinhlelo zokusebenza eziningi kwezobunjiniyela kanye nefiziksi. Ukuxoxa nge-integral eqondile akugcini nje ngokunikeza ukuqonda okuyisisekelo kwalomqondo kodwa futhi kuqinisa amakhono ethu okuhlaziya izibalo. Lesi sihloko sihlose ukunikeza izibonelo zezinkinga eziqondile eziqondile kanye nezingxoxo ezinemininingwane.
Umqondo Oyisisekelo Wokuhlanganiswa Okuqinisekile
Ngaphambi kokuthi singene ezinkingeni zesibonelo, ake sibukeze imiqondo eyisisekelo yama-integral aqondile. I-integral eqondile, ekhonjiswe yi-\(\int_a^bf(x) \, dx\), imele indawo engaphansi kwejika lomsebenzi \(f(x)\) kusukela ephuzwini \(x = a\) kuya ephuzwini \(x = b\).
Ngokwezibalo, i-integral eqondile kusukela ku-\(a\) kuya ku-\(b\) yomsebenzi \(f(x)\) ingachazwa kanje:
\[ \int_a^bf(x) \, dx = F(b) – F(a) \]
lapho \(F(x)\) kuyi-antiderivative ye-\(f(x)\).
Imibuzo Eyisibonelo Nengxoxo
Ake sibheke ezinye izibonelo zezinkinga eziqondile eziyisisekelo kanye nezingxoxo zazo.
Isibonelo Umbuzo 1
Umbuzo:
Bala i-integral eqondile yomsebenzi \(f(x) = 2x\) kusukela ku-\(x = 1\) kuya ku-\(x = 3\).
Ingxoxo:
Ukuze sixazulule le nhlanganisela, siqala ngokuthola i-antiderivative ye- \(f(x) = 2x\).
I-antiderivative ye- \(2x\) ithi:
\[ F(x) = x^2 + C \]
Kodwa-ke, kuma-integral aqondile asidingi i-constant of integration \(C\).
Manje, sebenzisa imikhawulo yama-integrals ukubala:
\[ \int_1^3 2x \, dx = F(3) – F(1) \]
Bala inani lika-\(F(x)\) kule mikhawulo:
\[ F(3) = 3^2 = 9 \]
\[ F(1) = 1^2 = 1 \]
Ngakho,
\[ \int_1^3 2x \, dx = 9 – 1 = 8 \]
Isibonelo Umbuzo 2
Umbuzo:
Bala i-integral eqondile yomsebenzi \(f(x) = x^2 + 1\) kusukela ku-\(x = 0\) kuya ku-\(x = 2\).
Ingxoxo:
Thola i-antiderivative ye- \(f(x) = x^2 + 1\).
I-antiderivative ye-\(x^2\) ithi:
\[ \frac{1}{3}x^3 \]
I-antiderivative ye-\(1\) ingu-\(x\).
Ngakho-ke, i-antiderivative ye- \(f(x)\) ithi:
\[ F(x) = \frac{1}{3}x^3 + x \]
Manje, sebenzisa imikhawulo yama-integrals ukubala:
\[ \int_0^2 (x^2 + 1) \, dx = F(2) – F(0) \]
Bala inani lika-\(F(x)\) kule mikhawulo:
\[ F(2) = \frac{1}{3}(2)^3 + 2 = \frac{8}{3} + 2 = \frac{8}{3} + \frac{6}{3} = \frac{14}{3} \]
\[ F(0) = \frac{1}{3}(0)^3 + 0 = 0 \]
Ngakho,
\[ \int_0^2 (x^2 + 1) \, dx = \frac{14}{3} – 0 = \frac{14}{3} \]
Isibonelo Umbuzo 3
Umbuzo:
Bala i-integral eqondile yomsebenzi \(f(x) = e^x\) kusukela ku-\(x = 1\) kuya ku-\(x = 2\).
Ingxoxo:
Thola i-antiderivative ye- \(f(x) = e^x\).
I-antiderivative ye-\(e^x\) ingu-\(e^x\).
Manje, sebenzisa imikhawulo yama-integrals ukubala:
\[ \int_1^2 e^x \, dx = F(2) – F(1) \]
Bala inani lika-\(F(x)\) kule mikhawulo:
\[ F(2) = e^2 \]
\[ F(1) = e^1 = e \]
Ngakho,
\[ \int_1^2 e^x \, dx = e^2 – e \]
Isibonelo Umbuzo 4
Umbuzo:
Bala i-integral eqondile yomsebenzi \(f(x) = \sin(x)\) kusukela ku-\(x = 0\) kuya ku-\(x = \pi\).
Ingxoxo:
Thola i-antiderivative ye- \(f(x) = \sin(x)\).
I-antiderivative ye-\(\sin(x)\) ingu-\(-\cos(x)\).
Manje, sebenzisa imikhawulo yama-integrals ukubala:
\[ \int_0^\pi \sin(x) \, dx = F(\pi) – F(0) \]
Bala inani lika-\(F(x)\) kule mikhawulo:
\[ F(\pi) = -\cos(\pi) = -(-1) = 1 \]
\[ F(0) = -\cos(0) = -1 \]
Ngakho,
\[ \int_0^\pi \sin(x) \, dx = 1 – (-1) = 1 + 1 = 2 \]
Isibonelo Umbuzo 5
Umbuzo:
Bala i-integral eqondile yomsebenzi \(f(x) = \frac{1}{x}\) kusukela ku-\(x = 1\) kuya ku-\(x = e\).
Ingxoxo:
Thola i-antiderivative ye- \(f(x) = \frac{1}{x}\).
I-antiderivative ye-\(\frac{1}{x}\) ingu-\(\ln|x|\).
Manje, sebenzisa imikhawulo yama-integrals ukubala:
\[ \int_1^e \frac{1}{x} \, dx = F(e) – F(1) \]
Bala inani lika-\(F(x)\) kule mikhawulo:
\[ F(e) = \ln(e) = 1 \]
\[ F(1) = \ln(1) = 0 \]
Ngakho,
\[ \int_1^e \frac{1}{x} \, dx = 1 – 0 = 1 \]
Isiphetho
Ngezibonelo ezingenhla, sizijwayeze ukuthola ama-integral aqondile emisebenzi ehlukahlukene eyisisekelo. Esinyathelweni ngasinye, kubalulekile ukuthola kuqala i-antiderivative bese usebenzisa imikhawulo ye-integral ukuthola inani lokugcina.
Ama-integral aqondile adlala indima ebalulekile emikhakheni eminingi yokufunda kanye nezicelo ezisebenzayo. Ukuqonda lo mqondo nokuzijwayeza ngezibonelo ezahlukahlukene kuzoqinisa kakhulu amakhono akho ezibalo.
