Fesili Fa'ata'ita'i ma Talanoaga i Meatotino o Fa'auigaga Mautinoa
O le integral mautinoa o se manatu faavae i le calculus, e matua aoga tele i le tele o faʻaoga i le matematika, fisiki, ma le inisinia. I totonu o lenei tusiga, o le a matou faʻamatalaina nisi o meatotino taua o le integral mautinoa ma tuʻuina atu faʻataʻitaʻiga ma fofo e faʻalolotoina ai lou malamalama i le autu.
Meatotino o Fa'auigaga Mautinoa
A o le’i o’o atu i fa’ata’ita’iga o fa’afitauli, se’i o tatou toe iloiloina nisi o meatotino fa’avae o integrals mautinoa e taua ona iloa:
1. Meatotino o le Linearity:
– Afai o \( f(x) \) ma le \( g(x) \) o ni galuega faatino e mafai ona tuufaatasia ma o \( a \) ma le \( b \) o ni tumau, ona:
\[
\int_a^b [af(x) + bg(x)] \, dx = a \int_a^bf(x) \, dx + b \int_a^bg(x) \, dx.
\]
2. Fa'aputuga o se Tumau:
– Afai o le \( c \) o se tumau, ona:
\[
\int_a^bc \, dx = c(b – a).
\]
3. Uiga o le Fa'aopoopoga o le Vaitaimi:
\[
\int_a^cf(x) \, dx + \int_c^bf(x) \, dx = \int_a^bf(x) \, dx
\]
4. Suiga o Tapula'a:
\[
\int_a^bf(x) \, dx = – \int_b^af(x) \, dx
\]
5. Zero i le tapula'a tutusa:
\[
\int_a^af(x) \, dx = 0
\]
Faʻataʻitaʻiga Fesili 1: Faʻaaogaina o le Meatotino Linearity
Faʻataʻitaʻiga o faʻafitauli:
Fuafua le tau o:
\[
\int_0^2 (3x^2 + 2x) \, dx
\]
Talanoaga:
Faaaoga le meatotino linearity e vaevae ai le integral i ni mea se lua:
\[
\int_0^2 (3x^2 + 2x) \, dx = \int_0^2 3x^2 \, dx + \int_0^2 2x \, dx
\]
Se'i o tatou fuafuaina le fa'aopoopoga muamua:
\[
\int_0^2 3x^2 \, dx
\]
\[
= 3 \int_0^2 x^2 \, dx
\]
\[
= 3 \left[ \frac{x^3}{3} \right]_0^2
\]
\[
= 3 \left( \frac{2^3}{3} – \frac{0^3}{3} \right)
\]
\[
= 3 \left( \frac{8}{3} \right)
\]
\[
= 8
\]
O lea la, tatou te fuafuaina le fa'aopoopoga lona lua:
\[
\int_0^2 2x \, dx
\]
\[
= 2 \int_0^2 x \, dx
\]
\[
= 2 \left[ \frac{x^2}{2} \right]_0^2
\]
\[
= 2 \left( 1 – 0 \right)
\]
\[
= 2
\]
Tuufaatasia ia taunuuga e lua:
\[
\int_0^2 (3x^2 + 2x) \, dx = 8 + 2 = 10
\]
Fa'ata'ita'iga Fesili 2: Fa'atasiga o se Tumau
Faʻataʻitaʻiga o faʻafitauli:
Fuafua le tau o:
\[
\int_1^4 5 \, dx
\]
Talanoaga:
I le faʻaaogaina o le meatotino taua o constants, e mafai ona tatou tusia:
\[
\int_1^4 5 \, dx = 5 \cdot (4 – 1)
\]
\[
= 5 \cdot 3
\]
\[
= 15
\]
Fa'ata'ita'iga Fesili 3: Uiga o le Suiga o Tapula'a
Faʻataʻitaʻiga o faʻafitauli:
Fa'amaonia e fa'apea:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx
\]
Talanoaga:
Tatou amata i le integral o le \( x^2 \) i luga o le va \( [2, 5] \):
\[
\int_2^5 x^2 \, dx = \left[ \frac{x^3}{3} \right]_2^5
\]
\[
= \frac{5^3}{3} – \frac{2^3}{3}
\]
\[
= \frac{125}{3} – \frac{8}{3}
\]
\[
= \frac{117}{3}
\]
\[
= 39
\]
Ia, se'i o tatou fa'atatauina le integral o le \( x^2 \) i luga o le va \( [5, 2] \) ma ia mautinoa e sui le fa'ailoga o le tali:
\[
\int_5^2 x^2 \, dx = \left[ \frac{x^3}{3} \right]_5^2
\]
\[
= \frac{2^3}{3} – \frac{5^3}{3}
\]
\[
= \frac{8}{3} – \frac{125}{3}
\]
\[
= -\frac{117}{3}
\]
\[
= -39
\]
Ua faʻamaonia e faʻapea:
\[
\int_2^5 x^2 \, dx = – \int_5^2 x^2 \, dx.
\]
Fa'ata'ita'iga Fesili 4: Uiga o le Fa'aopoopoga o le Vaitaimi
Faʻataʻitaʻiga o faʻafitauli:
Afai e iloa le \(\int_2^4 f(x) \, dx = 7\) ma le \(\int_4^6 f(x) \, dx = 5\), fuafua le tau o le \(\int_2^6 f(x) \, dx\).
Talanoaga:
Fa'aaogaina o le meatotino fa'aopoopo vaeluaga:
\[
2^6 f(x) \, dx = 2^4 f(x) \, dx + 4^6 f(x) \, dx
\]
\[
= 7 + 5
\]
\[
= 12
\]
I'uga
O le integral mautinoa e tele ona uiga taua e mafai ona fesoasoani ia i tatou e foia ituaiga eseese o faafitauli i se auala sili atu ona lelei. I totonu o lenei tusiga, ua matou talanoaina nisi o nei uiga faavae ma tuuina atu ni faataitaiga e faaalia ai le auala e mafai ai ona faaaogaina nei uiga i le faatinoga. Faatasi ai ma le malamalama ma le faataitai lelei, o le a mafai ona e foia faafitauli o integral mautinoa ma le mautinoa tele.
