Laʻana o nā nīnau kūkākūkā Riemann Sum
Pendahuluan
ʻO ka huina Riemann kahi manaʻo nui i ka calculus i hoʻohana ʻia e wehewehe i ka integral paʻa o kahi hana. Hoʻohana kēia ʻano hana i ka mahele waena a me ka huina o nā ʻāpana o nā huinahā e hoʻokokoke i ka integral. E kūkākūkā pono kēia ʻatikala i ke kumumanaʻo o ka huina Riemann, me nā laʻana a me nā kūkākūkā e hoʻomaʻalahi i ka hoʻomaopopo ʻana.
Manaʻo Kumu o ka Riemannian Sum
Ma mua o ko mākou kūkākūkā ʻana i nā hiʻohiʻona, he mea nui e hoʻomaopopo i ke kumumanaʻo kumu o nā huina Riemannian. Hiki ke hoʻokaʻawale ʻia nā huina Riemannian i ʻekolu mau ʻano nui:
1. Huina Riemann hema
2. Huina ʻākau ʻo Riemann
3. Huina Riemann waena
Hoʻokaʻawale kēia ʻano hana i ka manawa o ka hana e hoʻohui ʻia i loko o nā subintervals liʻiliʻi o ka lōʻihi like. A laila hoʻohana ʻia kēlā me kēia o kēia mau subintervals e hana i kahi huinahā nona ke kiʻekiʻe i hoʻoholo ʻia e ka waiwai o ka hana ma kahi kikoʻī i loko o ka subinterval (hema, ʻākau, a i ʻole waena).
ʻO ke ʻano maʻamau no ka Riemann Sum
Manaʻo mākou e makemake mākou e hoʻohui i ka hana \( f(x) \) mai \( a \) a i \( b \). Hoʻokaʻawale mākou i ka wā \( [a, b] \) i loko o \( n \) mau subintervals like o ka lōʻihi \( \Delta x = \frac{ba}{n} \). Hiki ke kākau ʻia nā huina Riemann no nā ʻano ʻekolu i ʻōlelo ʻia ma luna penei:
1. Hema ʻo Riemann:
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \]
2. ʻĀkau ʻo Riemann:
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \]
3. Riemann Waena:
\[ M_n = \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right) \Delta x \]
Ma hea:
– ʻO \( \Delta x \) ka laulā o kēlā me kēia subinterval.
– ʻO \( x_i \) ke kiko hoʻomaka o ka subinterval i-th no ka huina Riemann hema.
– ʻO \( x_i \) ka hopena o ka subinterval i-th no ka huina Riemann ʻākau.
– ʻO \( \frac{x_i + x_{i+1}}{2} \) ke kiko waena o ka subinterval i-th no ka huina Riemann waena.
Nā Nīnau Laʻana a me ke Kūkākūkā
E kūkākūkā kākou i nā pilikia hoʻohālike no kēlā me kēia ʻano o ka Riemann Sum e hoʻoikaika i ko kākou ʻike.
Laʻana 1: Huina Riemann Hema
E helu i ka huina Riemann hema no \( f(x) = x^2 \) ma ka wā \([0, 2]\) me \( n = 4 \).
Kūkākūkā:
1. Ka laulā o ka waena waena (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Wahi Māhele Waena (hema):
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5 \]
3. Ka Waiwai Hana ma ke Kiko Māhele:
\[ f(x_0) = f(0) = 0^2 = 0 \]
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. Hema 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 \]
\[ L_n = 0 + 0.125 + 0.5 + 1.125 \]
\[ L_n = 1.75 \]
Laʻana 2: Huina Riemann ʻĀkau
E helu i ka huina Riemann kūpono no \( f(x) = x^2 \) ma ka wā \([0, 2]\) me \( n = 4 \).
Kūkākūkā:
1. Ka laulā o ka waena waena (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Wahi Māhele Waena (ʻākau):
\[ x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, x_4 = 2.0 \]
3. Ka Waiwai Hana ma ke Kiko Māhele:
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. Huina ʻĀkau Riemann (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 \]
\[ R_n = 0.125 + 0.5 + 1.125 + 2 \]
\[ R_n = 3.75 \]
Laʻana 3: Huina Riemann Waena
E helu i ka huina Riemann waena no \( f(x) = x^2 \) ma ka wā \([0, 2]\) me \( n = 4 \).
Kūkākūkā:
1. Ka laulā o ka waena waena (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Waena o ka Subinterval:
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, \kikokikona{ a me } x_{n-1}=2.0 \]
Kikowaena waena o ka subinterval:
\[tm_0 = \left(\frac{0 + 0.5}{2}\right)=0.25 \]
\[tm_1 = \left(\frac{0.5 + 1.0}{2}\right)=0.75 \]
\[tm_2 = \left(\frac{1.0 + 1.5}{2}\right)=1.25 \]
\[tm_3 = \left(\frac{1.5 + 2.0}{2}\right)=1.75 \]
3. Ka Waiwai Hana ma ke Kikowaena:
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. Ka huina Riemann Waena (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 \]
\[ M_n = 0.03125 + 0.28125 + 0.78125 + 1.53125 \]
\[ M_n = 2.625 \]
Ka hopena
Ua kūkākūkā kēia ʻatikala pehea e helu ai i nā huina Riemann hema, ʻākau, a me waena, me nā laʻana kikoʻī. Hāʻawi ke ʻano huina Riemann i kahi ala kūpono e hoʻokokoke ai i ka integral o kahi hana ma ka puʻunaue ʻana i kona wā i nā subintervals liʻiliʻi a me ka helu ʻana i ka ʻāpana holoʻokoʻa o kēlā me kēia subinterval. He mea nui ka hoʻomaopopo maikaʻi ʻana i ka huina Riemann no ka poʻe e aʻo ana i ka calculus a i ʻole e hana ana me nā hana paʻakikī ma nā ʻano ʻepekema like ʻole.
