Ihe atụ nke ajụjụ mkparịta ụka Riemann Sum
Pendahuluan
Ngụkọta Riemann bụ echiche dị mkpa na mgbakọ na mwepụ nke ejiri kọwaa ihe dị mkpa nke ọrụ. Usoro a na-eji nkewa oge na mkpokọta mpaghara nke akụkụ anọ iji mee ka ihe dị mkpa dịkwuo mma. Isiokwu a ga-atụle echiche nke ngụkọta Riemann nke ọma, gụnyere ihe atụ na mkparịta ụka iji mee ka nghọta dị mfe.
Echiche bụ isi nke Riemannian Sum
Tupu anyị atụlee ihe atụ, ọ dị mkpa ịghọta echiche bụ isi nke ngụkọta Riemannian. Enwere ike kewaa ngụkọta Riemannian n'ụdị atọ bụ isi:
1. Ọnụọgụ Riemann aka ekpe
2. Ọnụ ego Riemann aka nri
3. Ọnụọgụ Riemann nke etiti
Usoro a na-agbaji oge ọrụ ahụ iji tinye ya na obere obere oge nke ogologo ya hà nhata. A na-eji obere oge ndị a nke ọ bụla n'ime ha emepụta akụkụ anọ nke uru ọrụ ahụ na-ekpebi elu ya site na isi ihe dị n'ime obere oge (aka ekpe, aka nri, ma ọ bụ etiti).
Usoro Izugbe maka Riemann Sum
Ka e were ya na anyị chọrọ ijikọta ọrụ \( f(x) \) site na \( a \) gaa na \( b \). Anyị kewara oge \( [a, b] \) n'ime obere oge nhata nke ogologo \( \Delta x = \frac{ba}{n} \). Enwere ike ide mkpokọta Riemann maka ụdị atọ ahụ a kpọtụrụ aha n'elu dị ka ndị a:
1. Riemann aka ekpe:
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \]
2. Riemann aka nri:
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \]
3. Etiti Riemann:
\[ M_n = \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\nri) \Delta x \]
Ebe:
– \( \Delta x \) bụ obosara nke obere oge ọ bụla.
– \( x_i \) bụ ebe mmalite nke obere oge i-th maka mkpụrụ Riemann aka ekpe.
– \( x_i \) bụ njedebe nke obere oge i-th maka ngụkọta Riemann ziri ezi.
– \( \frac{x_i + x_{i+1}}{2} \) bụ etiti nke obere oge i-th maka ngụkọta Riemann nke etiti.
Ajụjụ na Mkparịta ụka Ihe Nlereanya
Ka anyị tụlee ihe atụ nke nsogbu maka ụdị Riemann Sum ọ bụla iji mee ka nghọta anyị sikwuo ike.
Ihe atụ nke 1: Left Riemann Sum
Gbakọọ ngụkọta Riemann aka ekpe maka \( f(x) = x^2 \) na oge \([0, 2]\) na \(n = 4 \).
Azịza:
1. Obosara obere oge (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Ebe Nkewa Oge (aka ekpe):
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5 \]
3. Uru Ọrụ n'Ebe Nkewa:
\[ f(x_0) = f(0) = 0^2 = 0 \]
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
4. Aka ekpe Riemann Sum (Ln):
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x = (0) \cdot 0.5 + (0.25) \cdot 0.5 + (1) \cdot 0.5 + (2.25) \cdot 0.5 \]
\[ L_n = 0 + 0.125 + 0.5 + 1.125 \]
\[ L_n = 1.75 \]
Ihe atụ nke abụọ: Riemann Sum nke aka nri
Gbakọọ ngụkọta Riemann ziri ezi maka \( f(x) = x^2 \) na oge \([0, 2]\) na \( n = 4 \).
Azịza:
1. Obosara obere oge (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Ebe Nkewa Oge (n'aka nri):
\[ x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, x_4 = 2.0 \]
3. Uru Ọrụ n'Ebe Nkewa:
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
\[ f(x_4) = f(2.0) = (2.0)^2 = 4 \]
4. Riemann Sum (Rn) nke aka nri:
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x = (0.25) \cdot 0.5 + (1) \cdot 0.5 + (2.25) \cdot 0.5 + (4) \cdot 0.5 \]
\[ R_n = 0.125 + 0.5 + 1.125 + 2 \]
\[ R_n = 3.75 \]
Ihe atụ nke atọ: Middle Riemann Sum
Gbakọọ ngụkọta Riemann nke etiti maka \( f(x) = x^2 \) na oge \([0, 2]\) na \(n = 4 \).
Azịza:
1. Obosara obere oge (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Isi etiti nke obere oge:
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, \text{ na } x_{n-1}=2.0 \]
Isi etiti nke obere oge:
\[tm_0 = \left(\frac{0 + 0.5}{2}\right)=0.25 \]
\[tm_1 = \left(\frac{0.5 + 1.0}{2}\right)=0.75 \]
\[tm_2 = \left(\frac{1.0 + 1.5}{2}\right)=1.25 \]
\[tm_3 = \left(\frac{1.5 + 2.0}{2}\right)=1.75 \]
3. Uru Ọrụ na Midpoint:
\[ f(0.25) = (0.25)^2 = 0.0625 \]
\[ f(0.75) = (0.75)^2 = 0.5625 \]
\[ f(1.25) = (1.25)^2 = 1.5625 \]
\[ f(1.75) = (1.75)^2 = 3.0625 \]
4. Central Riemann Sum (Mn):
\[ M_n = \sum_{i=0}^{n-1} f(tm_i) \Delta x = (0.0625) \cdot 0.5 + (0.5625) \cdot 0.5 + (1.5625) \cdot 0.5 + (3.0625) \cdot 0.5 \]
\[ M_n = 0.03125 + 0.28125 + 0.78125 + 1.53125 \]
\[ M_n = 2.625 \]
Mmechi
Edemede a atụleela otu esi agbakọ ngụkọta aka ekpe, aka nri, na etiti Riemann, yana ihe atụ zuru ezu. Usoro nchikota Riemann na-enye ụzọ dị irè isi mee ka njikọ nke ọrụ dị mma site n'ikewa oge ya n'ime obere obere oge na ịgbakọ mpaghara mkpokọta nke obere oge ọ bụla. Nghọta dị mma nke ngụkọta Riemann dị mkpa maka ndị na-amụ calculus ma ọ bụ na-arụ ọrụ na ọrụ dị mgbagwoju anya na ngalaba sayensị dị iche iche.
