Cov Lus Nug Piv Txwv thiab Kev Sib Tham Txog Cov Khoom ntawm Cov Integrals Uas Paub Tseeb
Tus lej integral tseeb yog lub tswv yim tseem ceeb hauv kev xam lej, uas muaj txiaj ntsig zoo heev rau ntau yam kev siv hauv kev lej, physics, thiab engineering. Hauv tsab xov xwm no, peb yuav piav qhia txog qee yam tseem ceeb ntawm tus lej integral tseeb thiab muab cov piv txwv thiab cov lus teb los pab koj nkag siab tob txog cov ncauj lus.
Cov Khoom ntawm Cov Kev Sib Koom Ua Ke
Ua ntej peb nkag mus rau hauv cov teeb meem piv txwv, cia peb rov xyuas qee cov khoom yooj yim ntawm cov integrals uas tseem ceeb kom paub:
1. Cov Khoom Siv Linearity:
- Yog tias \( f(x) \) thiab \( g(x) \) yog cov functions uas sib xyaw tau thiab \( a \) thiab \( b \) yog cov constants, ces:
\[
\int_a^b [af(x) + bg(x)] \, dx = a \int_a^bf(x) \, dx + b \int_a^bg(x) \, dx.
\]
2. Kev Sib Koom Ua Ib Feem ntawm Ib Qho Tsis Hloov Pauv:
- Yog tias \(c\) yog qhov tsis hloov pauv, ces:
\[
\int_a^bc \, dx = c(b – a).
\]
3. Cov Khoom ntawm Kev Ntxiv Lub Sijhawm:
\[
\int_a^cf(x) \, dx + \int_c^bf(x) \, dx = \int_a^bf(x) \, dx
\]
4. Kev Hloov Pauv ntawm Cov Kev Txwv:
\[
\int_a^bf(x) \, dx = – \int_b^af(x) \, dx
\]
5. Xoom Ntawm Tib Qhov Txwv:
\[
\int_a^af(x) \, dx = 0
\]
Piv txwv lus nug 1: Siv cov khoom Linearity
Piv txwv teeb meem:
Xam tus nqi ntawm:
\[
\int_0^2 (3x^2 + 2x) \, dx
\]
Kev Sib Tham:
Siv cov cuab yeej linearity los cais cov integral ua ob:
\[
\int_0^2 (3x^2 + 2x) \, dx = \int_0^2 3x^2 \, dx + \int_0^2 2x \, dx
\]
Cia peb xam thawj qhov integral:
\[
\int_0^2 3x^2 \, dx
\]
\[
= 3 \int_0^2 x^2 \, dx
\]
\[
= 3 \left[ \frac{x^3}{3} \right]_0^2
\]
\[
= 3 \sab laug ( \frac{2^3}{3} – \frac{0^3}{3} \sab xis)
\]
\[
= 3 \sab laug ( \frac{8}{3} \sab xis)
\]
\[
= 8
\]
Tam sim no, peb xam qhov thib ob integral:
\[
\int_0^2 2x \, dx
\]
\[
= 2 \int_0^2 x \, dx
\]
\[
= 2 \left[ \frac{x^2}{2} \right]_0^2
\]
\[
= 2 \sab laug ( 1 – 0 \sab xis)
\]
\[
= 2
\]
Muab ob qho txiaj ntsig ua ke:
\[
\int_0^2 (3x^2 + 2x) \, dx = 8 + 2 = 10
\]
Piv txwv lus nug 2: Integral ntawm ib qho Constant
Piv txwv teeb meem:
Xam tus nqi ntawm:
\[
\int_1^4 5 \, dx
\]
Kev Sib Tham:
Siv cov khoom siv integral ntawm cov constants, peb tuaj yeem sau:
\[
\int_1^4 5 \, dx = 5 \cdot (4 – 1)
\]
\[
= 5 \cdot 3
\]
\[
= 15
\]
Piv txwv lus nug 3: Cov yam ntxwv ntawm kev hloov pauv txwv
Piv txwv teeb meem:
Ua pov thawj tias:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx
\]
Kev Sib Tham:
Peb pib nrog qhov sib xyaw ntawm \( x^2 \) ntawm qhov sib nrug \( [2, 5] \):
\[
\int_2^5 x^2 \, dx = \left[ \frac{x^3}{3} \right]_2^5
\]
\[
= \frac{5^3}{3} – \frac{2^3}{3}
\]
\[
= \frac{125}{3} – \frac{8}{3}
\]
\[
= \frac{117}{3}
\]
\[
= 39
\]
Tam sim no, cia peb xam qhov sib xyaw ntawm \( x^2 \) ntawm qhov sib nrug \( [5, 2] \) thiab xyuas kom tseeb tias rov qab lub cim ntawm cov lus teb:
\[
\int_5^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_5^2
\]
\[
= \frac{2^3}{3} – \frac{5^3}{3}
\]
\[
= \frac{8}{3} – \frac{125}{3}
\]
\[
= -\frac{117}{3}
\]
\[
= -39
\]
Nws tau ua pov thawj tias:
\[
\int_2^5 x^2 \, dx = - \int_5^2 x^2 \, dx.
\]
Piv txwv lus nug 4: Cov yam ntxwv ntawm kev ntxiv ntu
Piv txwv teeb meem:
Yog tias \(\int_2^4 f(x) \, dx = 7\) thiab \(\int_4^6 f(x) \, dx = 5\) paub lawm, xam tus nqi ntawm \(\int_2^6 f(x) \, dx\).
Kev Sib Tham:
Siv cov cuab yeej ntxiv ntawm lub sijhawm:
\[
\int_2^6 f(x) \, dx = \int_2^4 f(x) \, dx + \int_4^6 f(x) \, dx
\]
\[
= 7 + 5
\]
\[
= 12
\]
Xaus
Tus lej integral tseeb muaj ntau yam khoom tseem ceeb uas tuaj yeem pab peb daws ntau hom teeb meem kom zoo dua. Hauv tsab xov xwm no, peb tau tham txog qee yam khoom yooj yim no thiab muab piv txwv qhia txog yuav ua li cas siv cov khoom no hauv kev xyaum. Yog tias koj nkag siab thiab xyaum txaus, koj yuav muaj peev xwm daws cov teeb meem integral tseeb nrog kev ntseeg siab ntau dua.
