``Grin formulasi va uning tadbiqlari ``

Yüklə 1,19 Mb.

səhifə	4/8
tarix	25.03.2023
ölçüsü	1,19 Mb.
	#89754

1 2 3 4 5 6 7 8

kurs ishi mahliyo tayyor

B i rinc h i tur eg r i chizi q li int e g rall arni hisob la sh
1. 1- n ati j a.
1. 2- n ati j a.

Birinchi tur egri chiziqli integrallarning xossalari

Birinchi tur egri chiziqli integrallar ham Riman integrallari xossalari kabi xossalarga ega bo’ladi.Shuni e’tiborga olib , egri chiziqli integrallarning asosiy xossalarini sanab o’tish bilan kifoyalanamiz.

⁽¹^.4)^siste^m^a^bⁱ^l^aⁿ^ani^q^l^aⁿ^g^aⁿ^A^B^eg^r^{i c}^h^iz^iq^da^f^x^,^y^fuⁿ^ksi^y^a^b^erilg^aⁿ^va

uzluksiz.

1.Agar AB ACCB bo’lsa, u holda

^f^^x^,^y^d^s^^^f^^x^,^y^d^s^^^f^^x^,^y^d^sAB AC CB

bo’ladi.

2 .Ushbu

8
^c^f^^x^,^y^d^s^^c^f^^x^,^y^^^s^c^^cons^tAB AB

tenglik o’rinli.

^A^B^eg^rⁱ^c^hⁱ^z^iqda^f^x^,^y^f^u^nk^si^y^a^bil^aⁿ^g^x^,^y^f^un^k^si^y^a^ham^beri^l^g^aⁿ^va^u^z^l^u^ksiz.

3 .Quyidagi

^f^^x^,^y^^^^g^^x^,^y^^d^s^^^f^^x^,^y^^d^s^^^g^^x^,^y^d^sAB AB AB

formula o’rinli bo’ladi.

⁴^.^A^g^ar^^x^,^y^^A^B^da^f^x^,^y^⁰^bo^’l^sa^,^u^holda


^f^^x^,^y^d^s^⁰AB

bo’ladi.

⁵^.^f^x^,^y^f^u^nksi^y^{a shu}^A^B^daⁱ^nt^e^gr^a^ll^an^uvc^h^{i va}

^f^^x^,^y^^d^s^^^f^^x^,^y^^d^sAB AB

bo’ladi.

1
^{6 .}^Shunday^c^,^c₂^^A^B^nuq^ta^t^o^pil^a^diki^,



1
^f^^x^,^y^d^s^d^x^^^f^^c^,^c²^^^SAB

bo’ladi, bunda S AB ning uzunligi.Ushbu xossa o’rta qiymat haqidagi

teorema deb ataladi.

Birinchi tur egri chiziqli integrallarni hisoblash

Birinchi tur egri chiziqli integrallar, asosan Riman integrallariga keltirilib

hisoblanadi. Yuqorida keltirilgan 1.1-teoremaga ko’ra AB egri chiziq ushbu

_^x^^^x^s_^y^^^y^s

⁰^^^s^^^S

^siste^m^a^bil^an^b^er^il^g^a^nda^(b^u^nda^s^y^o^y^u^zunli^gⁱ⁾^ga^A^B^da^f^x^,^y^s^hu^uzlu^k^siz

bo’lganda egri chiziqli integral Riman integraliga keldi.Demak, bu Riman integralini hisoblash natijasida egri chiziqli integral topiladi. Endi AB egri chiziq ushbu

_^x^^^t_^y^^^^t

_⁰^^^s^^^S ₍₀_.₇₎

^sis^t^e^m^a^bilan⁽^pa^r^a^m^etrik^for^m^ada)^berilg^aⁿ^b^o^’lsin.^Bunda^^t^,^^^t^funksⁱ^y^alar^^,^^d^a^^'^t^,^^^'^t^ho^sⁱ^l^a^la^r^ga^e^ga^va^bu^hosil^a^lar^shu^ora^l^iqda^uz^l^uk^s^iz^ha^m^da^^^,^^^^^^A^v^a^^^,^^^^^^B^b^o^’lsin.

A A
^Ra^v^sh^aⁿ^ki^,¹^.⁷^si^s^te^m^a^^,^^o^r^al^iqni^A^B^egri^c^h^iz^iq^qa^ak^slanti^r^a^d^i.^B^unda^^,^^^^,^ⁿ^ing^A^B^eg^rⁱ^c^hⁱ^zⁱ^qd^agⁱ__{}__^aksiniⁿ^g^uzuⁿ^ligi




^^'²^t^^^'²^t^d^t

bo’ladi.


²^-t^e^ore^m^a.^A^g^ar^f^x^,^y^f^u^nk^sⁱ^y^a^AB^{da be}^r^ilg^aⁿ^v^a^uzl^u^ksiz^b^o^’lsa,^u^ho^l^da

^f^^x^,^y^^d^s^^^f^^^^t^^,^^^^t^^^^'2 ^^t^^^^^'2 ^^t^^d^tAB 

(0.8)

^bo’ladi. ^Isb^o^t^^,^^ora^liqnⁱ^ng

1 1
^P^^t₀^,^t^,^.^..^,^t_n^^^^t₀^^t^^.^.^.^^t_n ^^^

^b^o^’liⁿ^is^hⁱⁿⁱ^o^la^y^lik^.^Bu^bo’lin^is^hnⁱ^ng^bo^’^luv^chⁱ^nu^q^t^a^lari^t_k^k^⁰^,¹^,²^..^.^,ⁿⁿⁱ^ng^AB

_k k


_{dagi mos akslarni}^A^k^⁰^,¹^,²^..^.^,ⁿ_de_y^ladi._R^a^vs^ha^nkⁱ^,_bu^A^k^⁰^,¹^,²^...^,ⁿ^nuqtalar

^A^Begri chiziqning

A

0 1
^A^,^A^,^.^..^,_n

A

A
^b^o^’liⁿ^is^hⁱⁿⁱ^h^o^s^il^qiladi.^B^unda_k^^^t_k^,^^^t_k^k^⁰^,¹^,²^,^.^..^,ⁿ_v^a_k_k_₁ⁿⁱⁿ^guzunligi

t_k_₁s_k

_
t_k

^^'²^t^^^^'²^t^d^t

bo’ladi.O’rta qiymat haqidagi teoremadan foydalanib quyidagilarni topamiz:

^^s_k^^^^'²^_k^^^^'²^_k^^t_k_₁^^t_k^^^^'²^_k^^^^'²^_k^^^t_k^,

k k
^bunda^t_k^^t__^^t_k_₁^.Endi^^_k^^_k^,^^^_k^^_k^{deb ola}^m^iz^.^Ravshankⁱ^,^_k^,^_k^^A^A_₁^k^^⁰^,¹^,²^,^...^,ⁿ^¹^b^o’l^a^dⁱ^.^A^B^eg^rⁱ^c^hⁱ^z^iqnⁱ^ng^y^u^q^{orida a}^y^til^g^an

0 1 n
^A^,^A^,^.^..^,^A

^bo^’l_iⁿⁱ^shⁱⁿ_{i va h}_a_r^b_ir_k_k_₁^da^_k^,^_kⁿ^u^qt^aⁿⁱ^o^lⁱ^b,








n1
 f _k, _ks_kk0

yig’indini tuzamiz. Uni quyidagicha ham yozish mumkin:

^^^^n1 ^f^_k^,^_k^^^s_k^^n1 ^f^_k^,^^^_k^^'²^_k^^^^'²^_k^^^t_kk0 k0

(0.9)

^{Bu t}^e^ng_lⁱ^kⁿⁱⁿ^{g o}^’^{ng t}^o^m^onid^agⁱ^y^ig^’^indi^f^^t^,^^^t^^^^'²^t^^^^'²^t^^_funksiyaning^^,^^{oraliqdagi Rimat yig’indisidir.Shartga ko’ra}^f^x^,^y _va^^'^t^,^^^'^t_funksiyalar_{uzluksiz.Murakkab}^funksiyaning^{uzluksizligi haqidagi}_teore_m_adan^kelin_chiqadiki^us^hbu_funksiya^^,^^oraliqda_uzliksiz_._De_m_ak,^{bu funksiyalar}^^,^^oraliqda_{integralanuvchi}^bo’ladi_.

Ya’ni



lim


_m_a_x___t_k__₀^^n1 ^f^^^^^k^^,^^^^^k^^^^^'2 ^^^k^^^^^'2 ^^^k^^^t^k^^^f^^^^t^^,^^^^t^^^^^'2 ^^t^^^^^'2 ^^t^^d^t^Mod^o^m^ikⁱ^,^x^^^t^,^y^^^^t^funksi^y^a^l^ar^^,^^d^a^u^zl^u^ksiz^e^k^a^n,^u^nd^a^m^a^x^^t_k^⁰

dax_k0,y_k0 demak, s_k0. Bundan esa _p0 bo’lishi kelib chiqadi.(1.8)

munosabatdan foydalanib

_ⁱ_₀^^^^^f^^t^,^^^t^^'²^t^^^^'²^t^d^t

bo’lishini topamiz.Bu esa




^f^^x^,^y^d^s^^^f^^^^t^^,^^^^t^^^^^'2 ^^t^^^^^'2 ^^t^^d^tAB 

ekanini bildiradi.Teorema isbotlandi.

Bu teoramadan quyidagi natijalar kelib chiqadi.

1.1-natija. AB egri chiziq ushbu

^y^^^y^x^a^^^x^^b^,^y^a^^^A^,^y^b^^^B

^teⁿ^gla^m^a^bil^aⁿ^aⁿⁱ^q^l^aⁿ^g^an^bo’lⁱ^b,^y^^^y^x^fun^k^si^y^a^a^,^b^da^hosila^g^a^eg^a^va^uzluk^siz

bo’lsa, u holda

b


^f^^x^,^y^d^s^^^f^^x^,^y^^x^^^¹^^^y^'2 ^^x^^d^xAB a

bo’ladi.

1.2-natija. AB egri chiziq ushbu

^^^^^^^₀^^^^^₁


^t^e^ng^la^m^a^bilan^(q^u^tb^k^oo^rdⁱⁿ^a^ta^sⁱ^s^te^m^asida)^be^r^ilg^aⁿ^bo^’^li^b^,^^^f^uⁿ^k^si^y^a^₀^,^₁^da^h^osil^a^ga^e^ga^v^a^u^zluksiz^bo^’^lsin.Ag^a^r^f^x^,^y^f^u^nksi^y^a^shu^A^B^da^b^e^rilg^aⁿ^v^auzluksiz bo’lsa, u holda

_^f^^x^,^y^ds^^^f^^^co^s^^,^siⁿ^^^^2 ^^^^'2 ^d^^AB ₀

(0.10)

bo’ladi

Yüklə 1,19 Mb.

Dostları ilə paylaş:

1 2 3 4 5 6 7 8