Tusaalaha Su'aalaha Doodda Riemann Sum
Pendahuluan
Wadarta Riemann waa fikrad aasaasi ah oo ku jirta xisaabinta loo isticmaalo in lagu qeexo isku-dhafka qeexan ee shaqada. Habkani wuxuu adeegsadaa qaybinta kala-goynta iyo wadarta meelaha leydiyada si loo qiyaaso isku-dhafka. Maqaalkani wuxuu si fiican uga hadli doonaa fikradda wadarta Riemann, oo ay ku jiraan tusaalooyin iyo doodo si loo fududeeyo fahamka.
Fikradda Aasaasiga ah ee Riemannian Sum
Kahor inta aynaan ka hadlin tusaalooyinka, waxaa muhiim ah in la fahmo fikradda aasaasiga ah ee wadarta Riemannian. Wadarta Riemannian waxaa loo qaybin karaa saddex nooc oo waaweyn:
1. Wadarta Bidix ee Riemann
2. Wadarta Riemann ee midig
3. Wadarta Riemann ee bartamaha
Habkani wuxuu jebiyaa muddada shaqada si loogu daro waqtiyo hoose oo yar yar oo dhererkoodu siman yahay. Mid kasta oo ka mid ah muddooyinkan hoose waxaa markaa loo isticmaalaa in lagu sameeyo leydi dhererkiisu lagu go'aamiyo qiimaha shaqada meel gaar ah oo ku jirta muddo-hoosaadka hoose (bidix, midig, ama dhexe).
Qaacidada Guud ee Riemann Sum
Ka soo qaad inaan rabno inaan isku darno shaqada \( f(x) \) laga bilaabo \( a \) ilaa \( b \). Waxaan u qaybineynaa muddada u dhaxaysa \( [a, b] \) una qaybineynaa muddooyin hoose oo siman oo dherer ah \( \Delta x = \frac{ba}{n} \). Wadarta Riemann ee saddexda nooc ee kor ku xusan waxaa loo qori karaa sidan soo socota:
1. Bidix Riemann:
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x \]
2. Riemann Midig:
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x \]
3. Riemann Dhexe:
\[ M_n = \sum_{i=0}^{n-1} f\left(\frac{x_i + x_{i+1}}{2}\right) \Delta x \]
Halkee:
– \( \Delta x \) waa ballaca qayb-hoosaad kasta.
– \( x_i \) waa barta laga bilaabo muddada hoose ee i-th ee wadarta bidix ee Riemann.
– \( x_i \) waa dhammaadka muddada-hoosaadka i-th ee wadarta Riemann ee saxda ah.
– \( \frac{x_i + x_{i+1}}{2} \) waa barta dhexe ee i-th ee wadarta dhexe ee Riemann.
Su'aalo iyo Doodo Tusaale ah
Aan ka wada hadalno tusaale ahaan dhibaatooyinka nooc kasta oo Riemann Sum ah si aan u sii xoojinno fahamkeenna.
Tusaale 1: Bidix Riemann Sum
Xisaabi wadarta bidix ee Riemann ee \( f(x) = x^2 \) ee ku taal muddada \([0, 2]\) oo leh \( n = 4 \).
Dood:
1. Ballaca Dhex-dhexaadka Hoose (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Barta Kala Qaybinta Dhexda (bidix):
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5 \]
3. Qiimaha Shaqada ee Barta Qaybinta:
\[ f(x_0) = f(0) = 0^2 = 0 \]
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
4. Bidix Riemann Sum (Ln):
\[ L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x = (0) \cdot 0.5 + (0.25) \cdot 0.5 + (1) \cdot 0.5 + (2.25) \cdot 0.5 \]
\[ L_n = 0 + 0.125 + 0.5 + 1.125 \]
\[ L_n = 1.75 \]
Tusaale 2: Riemann Sum-ka Midig
Xisaabi wadarta Riemann ee saxda ah ee \( f(x) = x^2 \) ee ku jirta muddada \([0, 2]\) oo leh \( n = 4 \).
Dood:
1. Ballaca Dhex-dhexaadka Hoose (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Barta Kala Qaybinta Dhexda (midig):
\[ x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, x_4 = 2.0 \]
3. Qiimaha Shaqada ee Barta Qaybinta:
\[ f(x_1) = f(0.5) = (0.5)^2 = 0.25 \]
\[ f(x_2) = f(1.0) = (1.0)^2 = 1 \]
\[ f(x_3) = f(1.5) = (1.5)^2 = 2.25 \]
\[ f(x_4) = f(2.0) = (2.0)^2 = 4 \]
4. Riemann Sum (Rn):
\[ R_n = \sum_{i=1}^{n} f(x_i) \Delta x = (0.25) \cdot 0.5 + (1) \cdot 0.5 + (2.25) \cdot 0.5 + (4) \cdot 0.5 \]
\[ R_n = 0.125 + 0.5 + 1.125 + 2 \]
\[ R_n = 3.75 \]
Tusaale 3: Sum-ka Dhexe ee Riemann
Xisaabi wadarta dhexe ee Riemann ee \( f(x) = x^2 \) ee ku jirta muddada \([0, 2]\) oo leh \( n = 4 \).
Dood:
1. Ballaca Dhex-dhexaadka Hoose (Δx):
\[ \Delta x = \frac{ba}{n} = \frac{2-0}{4} = 0.5 \]
2. Barta dhexe ee bar-hoosaadka:
\[ x_0 = 0, x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, \qoraalka{ iyo } x_{n-1}=2.0 \]
Barta dhexe ee bar-hoosaadka:
\[tm_0 = \left(\frac{0 + 0.5}{2}\right)=0.25 \]
\[tm_1 = \left(\frac{0.5 + 1.0}{2}\right)=0.75 \]
\[tm_2 = \left(\frac{1.0 + 1.5}{2}\right)=1.25 \]
\[tm_3 = \left(\frac{1.5 + 2.0}{2}\right)=1.75 \]
3. Qiimaha Shaqada ee Bartamaha:
\[ f(0.25) = (0.25)^2 = 0.0625 \]
\[ f(0.75) = (0.75)^2 = 0.5625 \]
\[ f(1.25) = (1.25)^2 = 1.5625 \]
\[ f(1.75) = (1.75)^2 = 3.0625 \]
4. Bartamaha Riemann Sum (Mn):
\[ M_n = \sum_{i=0}^{n-1} f(tm_i) \Delta x = (0.0625) \cdot 0.5 + (0.5625) \cdot 0.5 + (1.5625) \cdot 0.5 + (3.0625) \cdot 0.5 \]
\[ M_n = 0.03125 + 0.28125 + 0.78125 + 1.53125 \]
\[ M_n = 2.625 \]
Gabagabo
Maqaalkani wuxuu ka hadlay sida loo xisaabiyo wadarta bidix, midig, iyo dhexe ee Riemann, iyo tusaalooyin faahfaahsan. Habka wadarta Riemann wuxuu bixiyaa hab wax ku ool ah oo lagu qiyaaso isku-dhafka shaqada iyadoo loo qaybinayo muddada u dhaxaysa qaybo yaryar oo yar yar isla markaana la xisaabinayo wadarta guud ee qayb kasta oo hoose. Faham wanaagsan oo ku saabsan wadarta Riemann waa lama huraan kuwa baranaya xisaabinta ama ka shaqeeya hawlaha adag ee qaybaha sayniska ee kala duwan.
