Nzira yeTrapezoidal muIntegrals
Mumasvomhu anoshandiswa, tinowanzosangana nedambudziko rekuverenga nzvimbo iri pasi pemugero. Mupfungwa, izvi zvinogona kugadziriswa uchishandisa ma integrals chaiwo. Zvisinei, mukuita, haasi ese mabasa anobatanidzwa zviri nyore nenzira yekuongorora. Mamwe mabasa akaoma, data rinowanikwa chete muchimiro chetafura, kana kuti mamodheru haana ma antiderivatives ari nyore. Mumamiriro ezvinhu akadai, nzira dzenhamba dzinova dzakakosha. Imwe yenzira dzinozivikanwa uye dzinoshandiswa kakawanda dzenhamba imutemo we trapezoidal, nzira yekufananidza ma integrals inoparadzanisa nzvimbo iri pasi pemugero kuita ma trapezoidal shapes.
Pfungwa Yekutanga yeDefinite Integral
Pachishandiswa geometriki, definite integral \(\int_a^bf(x)\,dx\) inogona kunzwisiswa senzvimbo yakaganhurirwa ne curve \(y=f(x)\), iyo \(x\) axis, uye mitsetse yakatwasuka \(x=a\) uye \(x=b\). Kana \(f(x)\ge 0\), iyo integral inzvimbo yakanaka. Kana basa racho riine negative values pane imwe nguva, iyo integral inopa "signed area".
Dambudziko guru rinomuka kana:
1. Basa \(f(x)\) harina mushonga unodzivirira unogona kuratidzwa nemabasa ekutanga.
2. Kukosha kwe \(f(x)\) kunongozivikanwa chete pane dzimwe nguva (data rekuyedza).
3. Kuverenga nekufananidzira hakushande zvakanaka kana kuti hakugoneki.
Apa ndipo panoshandiswa nzira ye trapezoidal kuyera kukosha kwekubatanidza nekushandisa ma function values pane dzimwe nzvimbo.
Pfungwa yeTrapezoidal Method
Nzira yetrapezoidal inotanga nepfungwa iri nyore: fungidzira curve ine mutsetse wakatwasuka pane imwe neimwe subinterval. Kunyange zvazvo nzira yeRiemann sum ichishandisa mativi akatwasuka nenzvimbo ine rectangle, nzira yetrapezoidal inoshandisa magumo maviri pane subinterval uye inoabatanidza nemutsetse wakatwasuka. Nzvimbo iri pasi pemutsetse wakatwasuka inoumba trapezoid, kwete rectangle.
Ngatitii tinoda kuverenga:
\[
\int_a^bf(x)\,dx
\]
Tinokamura interval \([a,b]\) kuita \(n\) subintervals dzine urefu hwakaenzana. Kureba kwe subinterval imwe neimwe ndekwekuti:
\[
h=\frac{ba}{n}
\]
Nzvimbo dzekuparadzanisa:
\[
x_0=a,\;x_1=a+h,\;x_2=a+2h,\;\madotsi,\;x_n=b
\]
Nemabasa akakosha:
\[
f(x_0), f(x_1), \madotsi, f(x_n)
\]
Pane imwe neimwe subinterval \([x_{i-1}, x_i]\), nzvimbo iri pasi pe curve inoyerwa nenzvimbo ye trapezoid:
\[
A_i \approx \frac{h}{2}\left[f(x_{i-1}) + f(x_i)\right]
\]
Zvadaro nzvimbo yese (integral approximation) inowanikwa nekuwedzera trapezoids dzese:
\[
\int_a^bf(x)\,dx \approx \sum_{i=1}^{n}\frac{h}{2}\left[f(x_{i-1}) + f(x_i)\right]
\]
Kana yakarongedzwa, fomura yemutemo wetrapezoidal inova:
\[
\int_a^b f(x)\,dx \approx \frac{h}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+\cdots+2f(x_{n-1})+f(x_n)\right]
\]
Cherechedza patani ye coefficient: mapoinzi ekupedzisira \(x_0\) uye \(x_n\) akawedzerwa ne1, nepo mapoinzi ari pakati akawedzerwa ne2.
Muenzaniso Wekuverenga Uri Nyore
Ngatitii tinoda kusvika:
\[
\int_0^2 x^2\,dx
\]
Pakuongorora, chinhu chakakosha ndeche \(\left.\frac{x^3}{3}\right|_0^2=\frac{8}{3}\approx 2{,}6667\). Zvisinei, tichashandisa nzira ye trapezoidal ne \(n=4\).
1. Nguva \([0,2]\), wobva:
\[
h=\frac{2-0}{4}=0{,}5
\]
2. Titik-titik: \(x_0=0\), \(x_1=0{,}5\), \(x_2=1\), \(x_3=1{,}5\), \(x_4=2\)
3. Kukosha kwebasa:
\(f(0)=0\)
\(f(0{,}5)=0{,}25\)
\(f(1)=1\)
\(f(1{,}5)=2{,}25\)
\(f(2)=4\)
Shandisa fomura ye trapezoid yakasanganiswa:
\[
T=\frac{h}{2}\left[f(x_0)+2f(x_1)+2f(x_2)+2f(x_3)+f(x_4)\right]
\]
\[
T=\frac{0{,}5}{2}\left[0+2(0{,}25)+2(1)+2(2{,}25)+4\right]
\]
\[
T=0{,}25\left[0{,}5+2+4{,}5+4\right]=0{,}25(11)=2{,}75
\]
Mhedzisiro inofungidzirwa ye2,75 iri pedyo zvikuru nemutengo chaiwo we2,6667, nemhosho yeinenge 0,0833.
Chikanganiso muTrapezoidal Method
Nzira yetrapezoidal inzira yekufungidzira, zvichireva kuti pane musiyano pakati pehuwandu hwekufungidzira nehukuru hwechokwadi hwekubatanidza. Hukuru hwekukanganisa hunokanganiswa ne:
1. Nhamba yezvikamu zvepakati \(n\): hombe \(n\), diki \(h\), uye kazhinji mhinduro dzacho dzakanyatsojeka.
2. Kukombama kwebasa: kana basa racho rakakombama zvikuru (rine derivative huru yechipiri), nzira yetrapezoidal inogona kunge isina kururama pane nzira dzemhando yepamusoro.
Kuti zvive nyore kushanda (zvine zvinobuda zvechipiri zvinoenderera mberi), kukanganisa kwenzira ye trapezoidal inogona kuverengerwa:
\[
E_T = -\frac{(ba)}{12}h^2 f”(\xi)
\]
kune dzimwe \(\xi\) pakati pe \(a\) na \(b\). Kubva mufomura iyi zvinogona kuonekwa kuti chikanganiso chinoenderana ne \(h^2\). Kureva kuti, kana tikapatsanura \(h\) nehafu (kureva, kuwedzera kaviri nhamba yezvikamu zvepakati), chikanganiso chinoderedzwa neinenge chikamu chimwe muzvina.
Mabhenefiti eTrapezoidal Method
Nzira ye trapezoidal inozivikanwa nekuda kwezvikonzero zvinoverengeka:
1. Zviri nyore uye zvinokurumidza
Kuverenga kuri nyore, uye kunogona kutoitwa nemaoko kuzvinhu zvidiki.
2. Yakakodzera data retafura
Kana kukosha kwe \(f(x)\) kuchingozivikanwa chete panzvimbo dzakasiyana (semuenzaniso mhedzisiro yekuyera), nzira iyi inogona kushandiswa zvakananga.
3. Kururama kuri nani pane nzira ye rectangular
Nekuti inoshandisa ma endpoint maviri uye inofananidza basa racho nemutsetse wakatwasuka, kazhinji inenge yakarurama kupfuura rectangular approximation yenhamba imwe chete yema partitions.
4. Yakagadzikana kune akawanda mainjiniya ekushandisa
Matambudziko mazhinji efizikisi neeinjiniya (semuenzaniso kuverenga basa, simba, nzvimbo inoyambuka, kuburitsa, nezvimwewo) anogona kugadziriswa nenzira iyi.
Zvipingamupinyi zveTrapezoidal Method
Kunyange zvazvo ichibatsira, nzira iyi ine miganhu:
1. Zvisina kunyatsojeka pamabasa akakombama zvakanyanya
Nekuti inoshandisa chete linear approximation pane imwe neimwe subinterval, mabasa ane shanduko dzinokurumidza kana sharp curves anoda \(n\ hombe).
2. Haisi nzira yepamusoro-soro
Zvikanganiso zve \(h^2\); dzimwe nzira dzakadai saSimpson dzinogona kupa zvikanganiso zvidiki pamusoro pehuwandu hwakafanana hwema subintervals (chero basa racho rakanyatsonaka).
3. Inonyanya kusarudzwa pakati pezvikamu
Kana chikamu chakafara zvakanyanya uye \(n\) chiri chidiki, mhedzisiro yacho inogona kusiyana zvakanyanya.
Penutup
Nzira yekubatanidza yetrapezoidal ndeimwe yenzira dzakakosha mukubatanidza nhamba. Nekupatsanura interval kuita zvikamu zvidiki uye kutsiva curve nemutsetse wakatwasuka pachikamu chimwe nechimwe, tinogona kuyera integral tisingatsvage antiderivative. Fomura iri nyore, iri nyore kushandisa, uye inonyanya kubatsira kana data rayo riri rega kana basa racho rakaoma kubatanidza.
Zvisinei, vashandisi venzira yetrapezoidal vanofanirwa kutarisisa huwandu hwema subintervals uye maitiro ebasa riri kubatanidzwa. Kune mabasa akakombama zvakanyanya kana kana zvichidikanwa kunyatsojeka, nzira yetrapezoidal inogona kushandiswa nekuwedzera \(n\), kana dzimwe nzira dzenhamba dzakadai senzira dzaSimpson kana Romberg dzinogona kufungwa. Nekunzwisisa kwakanaka, nzira yetrapezoidal inova chishandiso chinoshanda uye chinoshanda chekugadzirisa matambudziko akasiyana-siyana musainzi, mainjiniya, uye ehupfumi.
