Fa'ata'ita'iga o Fesili Talanoaga o le Riemann Sum
Pendahuluan
O le aofaʻi Riemann o se manatu autū i le calculus e faʻaaogaina e faʻamatalaina ai le integral mautinoa o se galuega faatino. O lenei metotia e faʻaaogaina ai le vaevaega o le vaeluaga ma le aofaʻi o vaega o faatafafā e faʻatatau i le integral. O lenei tusiga o le a talanoaina auiliili ai le manatu o le aofaʻi Riemann, e aofia ai faʻataʻitaʻiga ma talanoaga e faʻafaigofie ai le malamalama.
Manatu Fa'avae o le Riemannian Sum
A o le'i talanoaina ni fa'ata'ita'iga, e taua le malamalama i le manatu autū o aofa'iga Riemannian. E mafai ona vaevaeina aofa'iga Riemannian i ni ituaiga autū se tolu:
1. Aofa'iga tau'au o Riemann
2. Aofa'i o le Riemann taumatau
3. O le aofa'i o le ogatotonu o Riemann
O lenei metotia e motusia ai le va o le galuega faatino ina ia tuufaatasia i ni vaega laiti e tutusa le umi. Ona faaaogaina lea o nei vaega taitasi e fausia ai se faatafafā e fuafua lona maualuga e ala i le tau o le galuega faatino i se tulaga faapitoa i totonu o le vaega laiti (agavale, taumatau, po o le ogatotonu).
Fua Fa'atatau Lautele mo le Riemann Sum
Faapea tatou te mananao e tuufaatasia le galuega faatino \( f(x) \) mai \( a \) i le \( b \). Tatou te vaevaeina le vaeluaga \( [a, b] \) i \( n \) vaeluaga tutusa o le umi \( \Delta x = \frac{ba}{n} \). O aofaiga Riemann mo ituaiga e tolu o loʻo taʻua i luga e mafai ona tusia e pei ona taua i lalo:
1. Agavale Riemann:
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \]
2. Taumatau Riemann:
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \]
3. Riemann ogatotonu:
\[ M_n = \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right) \Delta x \]
O fea:
– \( \Delta x \) o le lautele o vaeluaga ta'itasi.
– \( x_i \) o le amataga lea o le i-th subinterval mo le aofa'i agavale o Riemann.
– \( x_i \) o le i'uga o le vaega lona i-th mo le aofa'i sa'o o Riemann.
– \( \frac{x_i + x_{i+1}}{2} \) o le ogatotonu lea o le vaega lona i mo le aofa'i ogatotonu o Riemann.
Fesili Fa'ata'ita'i ma Talanoaga
Seʻi o tatou talanoaina faʻataʻitaʻiga o faʻafitauli mo ituaiga taʻitasi o le Riemann Sum e faʻaloloto atili ai lo tatou malamalama.
Faʻataʻitaʻiga 1: Aofaʻi o Riemann Tauagavale
Fuafua le aofa'i agavale o Riemann mo le \( f(x) = x^2 \) i luga o le va \([0, 2]\) fa'atasi ai ma le \( n = 4 \).
Talanoaga:
1. Laʻutele o le vaeluaga i lalo (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Vaega Vaeluaina o le Vaitaimi (agavale):
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5 \]
3. Taua o Galuega Faatino i le Vaeluaga o le Nofoaga:
\[ f(x_0) = f(0) = 0^2 = 0 \]
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
4. Agavale Riemann Sum (Ln):
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x = (0) \cdot 0.5 + (0.25) \cdot 0.5 + (1) \cdot 0.5 + (2.25) \cdot 0.5 \]
\[ L_n = 0 + 0.125 + 0.5 + 1.125 \]
\[ L_n = 1.75 \]
Fa'ata'ita'iga 2: Taumatau Riemann Sum
Fuafua le aofa'i sa'o o Riemann mo le \( f(x) = x^2 \) i luga o le va \([0, 2]\) fa'atasi ai ma le \( n = 4 \).
Talanoaga:
1. Laʻutele o le vaeluaga i lalo (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Vaega Vaeluaina o le Vaitaimi (taumatau):
\[ x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, x_4 = 2.0 \]
3. Taua o Galuega Faatino i le Vaeluaga o le Nofoaga:
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
\[ f(x_4) = f(2.0) = (2.0)^2 = 4 \]
4. Taumatau Riemann Sum (Rn):
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x = (0.25) \cdot 0.5 + (1) \cdot 0.5 + (2.25) \cdot 0.5 + (4) \cdot 0.5 \]
\[ R_n = 0.125 + 0.5 + 1.125 + 2 \]
\[ R_n = 3.75 \]
Faʻataʻitaʻiga 3: Faʻaopoopoga o le Riemann i le ogatotonu
Fuafua le aofa'i ogatotonu o Riemann mo le \( f(x) = x^2 \) i luga o le vaeluaga \([0, 2]\) fa'atasi ai ma le \( n = 4 \).
Talanoaga:
1. Laʻutele o le vaeluaga i lalo (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Ogatotonu o le Vaeluaga:
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, \text{ ma } x_{n-1}=2.0 \]
Ogatotonu o le vaeluaga i lalo ifo o le vaeluaga:
\[tm_0 = \left(\frac{0 + 0.5}{2}\right)=0.25 \]
\[tm_1 = \left(\frac{0.5 + 1.0}{2}\right)=0.75 \]
\[tm_2 = \left(\frac{1.0 + 1.5}{2}\right)=1.25 \]
\[tm_3 = \left(\frac{1.5 + 2.0}{2}\right)=1.75 \]
3. Taua o Galuega i le Ogatotonu:
\[ f(0.25) = (0.25)^2 = 0.0625 \]
\[ f(0.75) = (0.75)^2 = 0.5625 \]
\[ f(1.25) = (1.25)^2 = 1.5625 \]
\[ f(1.75) = (1.75)^2 = 3.0625 \]
4. O le aofaʻi tutotonu o Riemann (Mn):
\[ M_n = \sum_{i=0}^{n-1} f(tm_i) \Delta x = (0.0625) \cdot 0.5 + (0.5625) \cdot 0.5 + (1.5625) \cdot 0.5 + (3.0625) \cdot 0.5 \]
\[ M_n = 0.03125 + 0.28125 + 0.78125 + 1.53125 \]
\[ M_n = 2.625 \]
I'uga
O lenei tusiga ua talanoaina ai le auala e fuafua ai le aofaʻi o Riemann agavale, taumatau, ma le ogatotonu, faʻatasi ai ma faʻataʻitaʻiga auiliili. O le metotia o le aofaʻi a Riemann e maua ai se auala lelei e faʻatatau ai le taua o se galuega faatino e ala i le vaevaeina o lona vaeluaga i ni vaeluaga laiti ma le fuafuaina o le aofaʻi atoa o vaeluaga taʻitasi. O se malamalama lelei i le aofaʻi a Riemann e taua tele mo i latou o loʻo suʻesuʻeina le calculus poʻo le galulue i galuega faatino faigata i vaega eseese o saienisi.
