Mohlala oa Lipotso tsa Puisano tsa Riemann Sum
Pendahuluan
Kakaretso ea Riemann ke khopolo ea motheo ka har'a lipalo e sebelisoang ho hlalosa karolo e tobileng ea mosebetsi. Mokhoa ona o sebelisa karohano ea nako le kakaretso ea libaka tsa likhutlonnetsepa ho hakanya karolo e kopaneng. Sehlooho sena se tla tšohla ka botlalo khopolo ea kakaretso ea Riemann, ho kenyeletsoa mehlala le lipuisano ho nolofatsa kutloisiso.
Khopolo-taba ea Motheo ea Riemannian Sum
Pele re buisana ka mehlala, ho bohlokoa ho utloisisa mohopolo oa motheo oa lipalo tsa Riemannian. Lipalo tsa Riemannian li ka aroloa ka mefuta e meraro e meholo:
1. Kakaretso e setseng ea Riemann
2. Kakaretso e nepahetseng ea Riemann
3. Kakaretso ea Riemann ea Bohareng
Mokhoa ona o arola karohano ea mosebetsi hore e kopanngoe le li-subinterval tse nyane tsa bolelele bo lekanang. E 'ngoe le e 'ngoe ea li-subinterval tsena e sebelisoa ho etsa khutlonnetsepa eo bophahamo ba eona bo khetholloang ke boleng ba mosebetsi ntlheng e itseng ka har'a subinterval (ka letsohong le letšehali, ka ho le letona, kapa bohareng).
Foromo e Akaretsang ea Riemann Sum
A re re re batla ho kopanya mosebetsi \( f(x) \) ho tloha \( a \) ho isa \( b \). Re arola karohano \( [a, b] \) ka \( n \) di-subinterval tse lekanang tsa bolelele \( \Delta x = \frac{ba}{n} \). Dipalopalo tsa Riemann tsa mefuta e meraro e boletsweng ka hodimo di ka ngolwa ka tsela e latelang:
1. Riemann ea Leqeleng:
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \]
2. Riemann ea ka letsohong le letona:
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \]
3. Riemann e Bohareng:
\[ M_n = \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right) \Delta x \]
Di mana:
– \( \Delta x \) ke bophara ba karolo e 'ngoe le e 'ngoe e ka tlase.
– \( x_i \) ke ntlha ya ho qala ha i-th subinterval bakeng sa kakaretso ya Riemann e letshehadi.
– \( x_i \) ke ntlha ya pheletso ya i-th subinterval bakeng sa kakaretso e nepahetseng ya Riemann.
– \( \frac{x_i + x_{i+1}}{2} \) ke ntlha e bohareng ya i-th subinterval bakeng sa kakaretso e bohareng ya Riemann.
Lipotso tsa Mehlala le Puisano
A re buisaneng ka mehlala ea mathata bakeng sa mofuta o mong le o mong oa Riemann Sum ho tebisa kutloisiso ea rona.
Mohlala oa 1: Kakaretso ea Riemann e Leqeleng
Bala kakaretso ea Riemann e letšehali bakeng sa \( f(x) = x^2 \) karolong ea \([0, 2]\) le \( n = 4 \).
Puisano:
1. Bophara ba Subinterval (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Sebaka sa ho Arola sa Nako (ka letsohong le letšehali):
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5 \]
3. Boleng ba Mosebetsi Sebakeng sa Karohano:
\[ f(x_0) = f(0) = 0^2 = 0 \]
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
4. Riemann Sum e ka Leqeleng (Lehlakoreng le ka Holimo):
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x = (0) \cdot 0.5 + (0.25) \cdot 0.5 + (1) \cdot 0.5 + (2.25) \cdot 0.5 \]
\[ L_n = 0 + 0.125 + 0.5 + 1.125 \]
\[ L_n = 1.75 \]
Mohlala oa 2: Kakaretso ea Riemann e nepahetseng
Bala kakaretso e nepahetseng ea Riemann bakeng sa \( f(x) = x^2 \) karolong ea \([0, 2]\) le \( n = 4 \).
Puisano:
1. Bophara ba Subinterval (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Sebaka sa ho Arola sa Nako (ka ho le letona):
\[ x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, x_4 = 2.0 \]
3. Boleng ba Mosebetsi Sebakeng sa Karohano:
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
\[ f(x_4) = f(2.0) = (2.0)^2 = 4 \]
4. Riemann Sum e nepahetseng (Rn):
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x = (0.25) \cdot 0.5 + (1) \cdot 0.5 + (2.25) \cdot 0.5 + (4) \cdot 0.5 \]
\[ R_n = 0.125 + 0.5 + 1.125 + 2 \]
\[ R_n = 3.75 \]
Mohlala oa 3: Kakaretso ea Riemann e Bohareng
Bala kakaretso e bohareng ea Riemann bakeng sa \( f(x) = x^2 \) karolong ea \([0, 2]\) le \( n = 4 \).
Puisano:
1. Bophara ba Subinterval (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Bohareng ba Subinterval:
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, \mongolo{ le } x_{n-1}=2.0 \]
Bohareng ba karolo e ka tlase:
\[tm_0 = \left(\frac{0 + 0.5}{2}\right)=0.25 \]
\[tm_1 = \left(\frac{0.5 + 1.0}{2}\right)=0.75 \]
\[tm_2 = \left(\frac{1.0 + 1.5}{2}\right)=1.25 \]
\[tm_3 = \left(\frac{1.5 + 2.0}{2}\right)=1.75 \]
3. Boleng ba Mosebetsi Bohareng:
\[ f(0.25) = (0.25)^2 = 0.0625 \]
\[ f(0.75) = (0.75)^2 = 0.5625 \]
\[ f(1.25) = (1.25)^2 = 1.5625 \]
\[ f(1.75) = (1.75)^2 = 3.0625 \]
4. Kakaretso ea Riemann Sum (Mn):
\[ M_n = \sum_{i=0}^{n-1} f(tm_i) \Delta x = (0.0625) \cdot 0.5 + (0.5625) \cdot 0.5 + (1.5625) \cdot 0.5 + (3.0625) \cdot 0.5 \]
\[ M_n = 0.03125 + 0.28125 + 0.78125 + 1.53125 \]
\[ M_n = 2.625 \]
Qetello
Sengoloa sena se buile ka mokhoa oa ho bala lipalo tsa Riemann tse letšehali, tse letona le tse bohareng, hammoho le mehlala e qaqileng. Mokhoa oa kakaretso oa Riemann o fana ka mokhoa o atlehang oa ho hakanya karolo ea bohlokoa ea mosebetsi ka ho arola karohano ea oona ka li-subinterval tse nyane le ho bala sebaka sohle sa subinterval ka 'ngoe. Kutloisiso e ntle ea kakaretso ea Riemann e bohlokoa bakeng sa ba ithutang calculus kapa ba sebetsang ka mesebetsi e rarahaneng masimong a fapaneng a saense.
