ʻO Riemann Sum: Kekahi o nā pou o ka Calculus Integral
I ka makemakika, ʻoi aku hoʻi i ka calculus integral, he kuleana koʻikoʻi ke kumumanaʻo o ka huina Riemann. Ua hoʻolauna ʻia e ka makemakika Kelemania kaulana ʻo Bernhard Riemann, ʻo kēia ʻano hana he ala koʻikoʻi ia e wehewehe ai i ka hoʻohui ʻana o kahi hana ma luna o kahi manawa i hāʻawi ʻia. ʻO ka hoʻomaopopo ʻana i ka huina Riemann e hiki ai iā mākou ke kuhi i ka wahi ma lalo o kahi piʻo, kahi noi koʻikoʻi ma nā ʻano ʻepekema a me ka ʻenehana he nui, mai ka physics a i ka hoʻokele waiwai.
I mea e hoʻomaopopo ai i ke ʻano o ka huina Riemann, pono mākou e nānā i kāna mau mea kumu, me ka hoʻokaʻawale ʻana i ka wā, ka hoʻoholo ʻana i nā kiko loiloi, ke kūkulu ʻana i nā huina, a me kā lākou noi ʻana i ka hoʻohui ʻana. E ʻeli hohonu kākou i kēia kumuhana.
Hoʻolauna i nā Manaʻo Kumu
ʻO ka huina Riemann kahi ala e helu ai i ka hoʻohui paʻa o kahi hana ma luna o kahi wā pani \([a, b]\). Pili kēia ʻano hana i ka hoʻokaʻawale ʻana i ka wā i nā subintervals liʻiliʻi, e loiloi ana i ka hana ma nā wahi kikoʻī i kēlā me kēia subinterval, a laila e hōʻuluʻulu i nā huahana o nā waiwai hana me nā lōʻihi o nā subintervals pili.
Māhele Waena
ʻO ka hana mua i ka wehewehe ʻana i ka huina Riemann, ʻo ia ke māhele ʻana i ka wā \([a, b]\) i nā subintervals o kahi lōʻihi i hāʻawi ʻia. Manaʻo ʻia ua māhele ʻia ka wā \([a, b]\) i nā ʻāpana like \(n\), a laila:
\[ \Delta x = \frac{b – a}{n} \]
Loaʻa i kēlā me kēia subinterval ka lōʻihi \(\Delta x\), a ʻo kēia mau kiko mahele maʻamau ʻo \((x_0, x_1, x_2, …, x_n)\), kahi \(x_0 = a\), \(x_1 = a + \Delta x\), \(x_2 = a + 2\Delta x\), a pēlā aku a hiki i \(x_n = b\).
Ka Hoʻoholo ʻana o ke Kiko Loiloi
No kēlā me kēia subinterval \([x_{i-1}, x_i]\), pono kahi kiko loiloi \(x_i \) e waiho ana i loko o kēlā subinterval. Hiki ke hoʻoholo ʻia kēia kiko penei:
1. Kiko Hema: \(x_i^ = x_{i-1}\)
2. Kiko ʻĀkau: \(x_i^ = x_i\)
3. Kikowaena: \(x_i^ = \frac{x_{i-1} + x_i}{2}\)
4. Nā Kiko Kaulike: ʻO kēlā me kēia \(x_i \) he kiko kaulike ia ma \([x_{i-1}, x_i]\)
Hiki i ke koho ʻana o nā kiko loiloi ke hoʻopili i ka hopena o ka huina Riemann, ʻoiai inā he hana hoʻomau ʻole a loli wikiwiki paha.
Ka Hoʻokumu ʻana o ka Sum
Ke pau ka hoʻokaʻawale ʻana o ka manawa a me ka hoʻoholo ʻana i ke kiko loiloi, ʻo ka hana aʻe, ʻo ia ka helu ʻana i ka waiwai hana ma kēlā me kēia kiko loiloi \(f(x_i^ )\) a hoʻonui i kēlā waiwai me ka lōʻihi subinterval \(\Delta x\). Ua wehewehe ʻia ka huina Riemann \(R\) penei:
\[ R = \sum_{i=1}^nf(x_i^ ) \Delta x \]
Ke hoʻonui ʻia ka helu o nā subintervals \(n\) me ka ʻole o ka palena (\(n \rightarrow \infty\)), lilo ka lōʻihi o ka subinterval \(\Delta x\) i mea liʻiliʻi loa a hoʻokokoke ka huina Riemann i ka integral o ka hana \(f\) ma ka interval \([a, b]\). Ua kākau ʻia kēia palena palena penei:
\[ \int_a^bf(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^nf(x_i^ ) \Delta x \]
Laʻana Hoʻokō o Riemann Sum
Ma ke ʻano he hoʻākāka, e hoʻopili kākou i ka huina Riemann e hoʻoholo ai i ka integral o ka hana \(f(x) = x^2\) ma ka wā \([0, 1]\).
KaʻAnuʻu Hana 1: Māhele Waena
Manaʻo mākou e puʻunaue kākou i ka wā \([0, 1]\) i nā \(n\) subintervals o ka lōʻihi like, a laila ʻo ka lōʻihi o nā subintervals:
\[ \Delta x = \frac{1 – 0}{n} = \frac{1}{n} \]
KaʻAnuʻu Hana 2: Wahi Loiloi
E hoʻohana i ke kiko waena \(x_i \) e loiloi i ka hana ma kēlā me kēia subinterval \([x_{i-1}, x_i]\):
\[ x_i^ = \frac{x_{i-1} + x_i}{2} = \frac{\left(\frac{i-1}{n}\ʻākau) + \left(\frac{i}{n}\ʻākau)}{2} = \frac{2i – 1}{2n} \]
KaʻAnuʻu Hana 3: E helu i ka Huina
ʻO ka waiwai o ka hana \(f(x_i^) = \left( \frac{2i – 1}{2n} \right)^2 = \frac{(2i-1)^2}{4n^2}\), a laila lilo ka huina Riemann i:
\[ R = \sum_{i=1}^nf\left(\frac{2i – 1}{2n}\right) \Delta x = \sum_{i=1}^n \frac{(2i-1)^2}{4n^2} \cdot \frac{1}{n} = \frac{1}{4n^3} \sum_{i=1}^n (2i-1)^2 \]
Me ka loiloi hou aku, hāʻawi ka huina o nā huinaha o nā helu ʻē i ka hōʻailona sigma i hiki ke hoʻomaʻalahi hou ʻia a hiki i ka palena.
ʻO ka mea hope loa, i ka hele ʻana o \(n\) i ka palena ʻole, e hoʻokokoke aku ka waiwai o ka huina Riemann i ka hopena o ka integral pololei:
\[ \lim_{n \to \infty} \frac{1}{4n^3} \sum_{i=1}^n (2i-1)^2 = \frac{1}{3} \]
A ma ka hopena analytical o ka integral loaʻa iā mākou:
\[ \int_0^1 x^2 \, dx = \left[ \frac{x^3}{3} \right]_0^1 = \frac{1}{3} \]
Nā ʻano like ʻole a me nā noi o nā Riemann Sums
Ma waho aʻe o ka hoʻohui kuʻuna, loaʻa pū kekahi mau ʻano like ʻole o ka huina Riemann, me ka huina Riemann-Kronecker a me ka huina Riemann-Stieltjes no nā wahi metric a me nā noi ākea i ka loiloi hana. Hoʻokumu pū ia i ke kumu no nā ʻano helu e like me nā ʻano Trapzoid a me Simpson i hoʻohana ʻia i ka helu ʻepekema.
Pani
Hāʻawi nā huina Riemann i kahi ʻano hana paʻa a maʻalahi hoʻi no ka wehewehe ʻana a me ka helu ʻana i nā integrals i nā ʻano makemakika like ʻole. Ma ke ʻano he mea hana aʻo no nā pilikia integral kumu i ka calculus, ʻo ka hoʻomaopopo piha ʻana i kēia manaʻo e wehe i nā ʻike i nā noi ākea o nā integrals i ke ola maoli, ma nā ʻepekema kikoʻī a me nā ʻāpana socio-economic. ʻAʻole wale ʻo Bernhard Riemann i hoʻonui i ke kumumanaʻo makemakika me kēia ʻike akā ua wehe pū i nā ala hou i ka loiloi integral hou.
