Muenzaniso weMibvunzo yeKukurukurirana yeRiemann Sum
Pendauluan
Mari yeRiemann ipfungwa huru mukuverenga inoshandiswa kutsanangura chinhu chakakosha chebasa. Nzira iyi inoshandisa kupatsanurana kwepakati nenhamba yenzvimbo dzema rectangles kuti ifungidzire chinhu chakakosha. Chinyorwa chino chichakurukura zvakadzama pfungwa yeMari yeRiemann, kusanganisira mienzaniso nehurukuro kuti zviite kuti vanhu vanzwisise.
Pfungwa huru yeRiemannian Sum
Tisati takurukura mienzaniso, zvakakosha kunzwisisa pfungwa huru yehuwandu hweRiemannian. Huwandu hweRiemannian hunogona kukamurwa kuita mhando nhatu huru:
1. Huwandu hweRiemann hwakasara
2. Kurudyi Riemann uwandu
3. Nhamba yepakati yeRiemann
Nzira iyi inoputsa nguva yebasa kuti ibatanidzwe muzvikamu zvidiki zvehurefu hwakaenzana. Imwe neimwe yezvikamu izvi zvehurefu inoshandiswa kugadzira rectangle ine kukwirira kwayo kunotemerwa nekukosha kwebasa pane imwe nzvimbo mukati mezvikamu zvehurefu (kuruboshwe, kurudyi, kana pakati).
Fomura Yakazara yeRiemann Sum
Ngatitii tinoda kubatanidza basa \( f(x) \) kubva \( a \) kusvika \( b \). Tinokamura interval \( [a, b] \) kuita \( n \) subintervals dzakaenzana dze length \( \Delta x = \frac{ba}{n} \). Mari yeRiemann yemhando nhatu dzataurwa pamusoro apa inogona kunyorwa seinotevera:
1. Riemann kuruboshwe:
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \]
2. Kurudyi Riemann:
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \]
3. Middle Riemann:
\[ M_n = \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right) \Delta x \]
Di mana:
– \( \Delta x \) upamhi hwechikamu chimwe nechimwe chepakati penguva.
– \( x_i \) ndiyo nzvimbo inotangira i-th subinterval yeRiemann sum yekuruboshwe.
– \( x_i \) ndiyo magumo e i-th subinterval yeRiemann sum yekurudyi.
– \( \frac{x_i + x_{i+1}}{2} \) ndiyo nzvimbo iri pakati pei-th subinterval yepakati peRiemann sum.
Mibvunzo yemuenzaniso nekukurukurirana
Ngatikurukurei mienzaniso yezvinetso zverudzi rumwe nerumwe rweRiemann Sum kuti tinzwisise zvakadzama.
Muenzaniso 1: Rutivi rweRiemann rweruboshwe
Verenga huwandu hweRiemann hwekuruboshwe hwe \( f(x) = x^2 \) pane imwe nguva \([0, 2]\) na \( n = 4 \).
Kukurukurirana:
1. Upamhi hweSubing Interval (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Nzvimbo Yekupatsanurana Pakati (kuruboshwe):
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5 \]
3. Kukosha kwebasa paDividing Point:
\[ f(x_0) = f(0) = 0^2 = 0 \]
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
4. Kuruboshwe Riemann Sum (Kurudyi):
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x = (0) \cdot 0.5 + (0.25) \cdot 0.5 + (1) \cdot 0. 5 + (2.25) \cdot 0.5 \]
\[ L_n = 0 + 0.125 + 0.5 + 1.125 \]
\[ L_n = 1.75 \]
Muenzaniso 2: Right Riemann Sum
Verenga huwandu hwakakodzera hweRiemann hwe \( f(x) = x^2 \) pane imwe nguva \([0, 2]\) na \( n = 4 \).
Kukurukurirana:
1. Upamhi hweSubing Interval (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Nzvimbo Yekupatsanurana Pakati (kurudyi):
\[ x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, x_4 = 2.0 \]
3. Kukosha kwebasa paDividing Point:
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
\[ f(x_4) = f(2.0) = (2.0)^2 = 4 \]
4. Kurudyi Riemann Sum (Rn):
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x = (0.25) \cdot 0.5 + (1) \cdot 0.5 + (2.25) \cdot 0.5 + (4) \cdot 0.5 \]
\[ R_n = 0.125 + 0.5 + 1.125 + 2 \]
\[ R_n = 3.75 \]
Muenzaniso 3: Middle Riemann Sum
Verenga huwandu hwepakati hweRiemann hwe \( f(x) = x^2 \) pane imwe nguva \([0, 2]\) na \( n = 4 \).
Kukurukurirana:
1. Upamhi hweSubing Interval (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Pakati peSubinterval:
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, \text{ na } x_{n-1}=2.0 \]
Pakati pechikamu chepakati:
\[tm_0 = \kuruboshwe(\frac{0 + 0.5}{2}\kurudyi)=0.25 \]
\[tm_1 = \kuruboshwe(\frac{0.5 + 1.0}{2}\kurudyi)=0.75 \]
\[tm_2 = \kuruboshwe(\frac{1.0 + 1.5}{2}\kurudyi)=1.25 \]
\[tm_3 = \kuruboshwe(\frac{1.5 + 2.0}{2}\kurudyi)=1.75 \]
3. Kukosha kwebasa paMidpoint:
\[ f(0.25) = (0.25)^2 = 0.0625 \]
\[ f(0.75) = (0.75)^2 = 0.5625 \]
\[ f(1.25) = (1.25)^2 = 1.5625 \]
\[ f(1.75) = (1.75)^2 = 3.0625 \]
4. Central Riemann Sum (Mn):
\[ M_n = \sum_{i=0}^{n-1} f(tm_i) \Delta x = (0.0625) \cdot 0.5 + (0.5625) \cdot 0.5 + (1.5625) \cdot 0.5 + (3.0625) \cdot 0.5 \]
\[ M_n = 0.03125 + 0.28125 + 0.78125 + 1.53125 \]
\[ M_n = 2.625 \]
Mhedziso
Chinyorwa chino chakurukura maverengero eRiemann sums kuruboshwe, kurudyi, uye pakati, pamwe chete nemienzaniso yakadzama. Nzira yeRiemann sum inopa nzira inoshanda yekufungidzira integral yebasa nekukamura interval yaro kuita subintervals diki uye kuverenga nzvimbo yese ye subinterval yega yega. Kunzwisisa kwakanaka kweRiemann sum kwakakosha kune avo vanodzidza calculus kana vanoshanda nemabasa akaomarara muminda yakasiyana-siyana yesainzi.
