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Ôîðìóëà (9.2.5) âèðàæຠåêîíîì³÷íèé çì³ñò âèçíà÷åíîãî<br />
³íòåãðàëà.<br />
Ìîæíà ðîçãëÿíóòè ùå íèçêó çàäà÷ çì³ñòîâíîãî õàðàêòåðó,<br />
ùî ïðèâîäÿòü äî ïîíÿòòÿ âèçíà÷åíîãî ³íòåãðàëà, àëå<br />
òðåáà â÷àñíî çóïèíèòèñÿ ³ ïåðåéòè äî ôîðìóëþâàíü äåÿêèõ<br />
óìîâ ³íòåãðîâíîñò³ ôóíêö³¿ f(õ) íà ñåãìåíò³ [a, b].<br />
Òåîðåìà 9.2.1 (ïðî íåîáõ³äíó óìîâó ³íòåãðîâíîñò³).<br />
ßêùî ôóíêö³ÿ ó = f(õ) ³íòåãðîâíà íà ñåãìåíò³ [à, b], òî âîíà<br />
îáìåæåíà íà íüîìó.<br />
Ä î â å ä å í í ÿ. Ïðèïóñòèìî ïðîòèëåæíå, òîáòî, ùî ôóíêö³ÿ<br />
íåîáìåæåíà íà â³äð³çêó [à, b]. Òîä³ äëÿ áóäü-ÿêîãî ðîçáèòòÿ<br />
R ñåãìåíòà [a, b] íà ÷àñòèíí³ ôóíêö³ÿ f(õ) áóäå íåîáìåæåíîþ<br />
õî÷à á íà îäíîìó ç íèõ, íàïðèêëàä, äëÿ êîíêðåòíîñò³<br />
íà [0, õ 1 ] (íà ðåøò³ ÷àñòèííèõ ñåãìåíò³â ôóíêö³ÿ f(õ)<br />
ââàæàºòüñÿ îáìåæåíîþ). Òåïåð ³íòåãðàëüíó ñóìó, ÿêó ìè<br />
ïîçíà÷èìî ÷åðåç S n , ìîæíà çîáðàçèòè ó âèãëÿä³:<br />
S n = f(ξ 1 )∆x 1 + σ n ,<br />
n<br />
äå ∑ f( )<br />
σ = ξ ∆x.<br />
n i i<br />
i=<br />
2<br />
Ïîñë³äîâí³ñòü σ n çã³äíî ç ïðèïóùåííÿì — îáìåæåíà, à<br />
ïåðøèé äîäàíîê ñóìè çà ðàõóíîê íåîáìåæåíîñò³ ôóíêö³¿<br />
f(õ) íà [0, õ 1 ] ìîæíà çðîáèòè òàêèì âåëèêèì, ùî çà àáñîëþòíîþ<br />
âåëè÷èíîþ â³í áóäå ïåðåâåðøóâàòè ÿêå çàâãîäíî<br />
çàäàíå äîäàòíå ÷èñëî. À öå îçíà÷àº, ùî ãðàíèöÿ ³íòåãðàëüíî¿<br />
ñóìè S n íå ³ñíóº ïðè ρ→ 0 , à òîä³ ôóíêö³ÿ f(õ) íå º ³íòåãðîâíîþ,<br />
ùî ñóïåðå÷èòü óìîâ³ òåîðåìè. Òåîðåìó äîâåäåíî.<br />
Çàóâàæåííÿ. Îáåðíåíå òâåðäæåííÿ íåïðàâèëüíå, òîáòî<br />
ç îáìåæåíîñò³ ôóíêö³¿ íà cåãìåíò³ íå âèïëèâຠ¿¿ ³íòåãðîâí³ñòü<br />
íà íüîìó. Êëàñè÷íèì ïðèêëàäîì òàêî¿ ôóíêö³¿ º òàê<br />
çâàíà ôóíêö³ÿ ijð³õëå (äèâ. ï. 6.1.2). Öÿ ôóíêö³ÿ íà â³äð³çêó<br />
[0, 1] º îáìåæåíîþ, òîìó ùî 0 ≤ D(õ) ≤ 1. Äîâåäåìî, ùî<br />
âîíà íå ³íòåãðîâíà íà [0, 1]. Ðîç³á’ºìî â³äð³çîê [0, 1] äîâ³ëüíèì<br />
÷èíîì íà ÷àñòèíí³ ñåãìåíòè ³ ñêëàäåìî ³íòåãðàëüíó<br />
ñóìó<br />
n<br />
n<br />
= ∑ ( ξi)<br />
∆<br />
i<br />
i=<br />
1<br />
S D x .<br />
Ðîçãëÿíåìî òåïåð äâà âèïàäêè âèáîðó òî÷îê ξ i :<br />
1) òî÷êè ξ i — ðàö³îíàëüí³; 2) òî÷êè ξ i — ³ððàö³îíàëüí³.<br />
Ó âèïàäêó 1) ³íòåãðàëüíà ñóìà<br />
ó âèïàäêó 2) ³íòåãðàëüíà ñóìà<br />
S<br />
n<br />
S<br />
n<br />
n<br />
i=<br />
1<br />
n<br />
= ∑ 1⋅∆ x = 1 ³<br />
= ∑ 0⋅∆ x = 0 ³<br />
i=<br />
1<br />
i<br />
i<br />
limS<br />
n<br />
= 1, à<br />
ρ→0<br />
limS<br />
n<br />
= 0.<br />
ρ→0<br />
Îòæå, ãðàíèöÿ ³íòåãðàëüíî¿ ñóìè çàëåæèòü â³ä âèáîðó<br />
òî÷îê ξ i , à öå îçíà÷àº, ùî ôóíêö³ÿ D(õ) íå º ³íòåãðîâíîþ íà<br />
[0, 1].<br />
Òåîðåìà 9.2.2. (ïðî äîñòàòíþ óìîâó ³íòåãðîâíîñò³).<br />
ßêùî ôóíêö³ÿ f(õ) íåïåðåðâíà íà ñåãìåíò³ [à, b], òî âîíà<br />
³íòåãðîâíà íà íüîìó.<br />
Òåîðåìà 9.2.3. ßêùî ôóíêö³ÿ f(õ) îáìåæåíà íà ñåãìåíò³<br />
[a, b] ³ íåïåðåðâíà íà íüîìó óñþäè, êð³ì ñê³í÷åííîãî<br />
÷èñëà òî÷îê, â ÿêèõ ìຠòî÷êè ðîçðèâó I ðîäó, òî âîíà ³íòåãðîâíà<br />
íà íüîìó.<br />
Òåîðåìà 9.2.4. Âñÿêà îáìåæåíà ³ ìîíîòîííà íà ñåãìåíò³<br />
[à, b] ôóíêö³ÿ ³íòåãðîâíà íà íüîìó.<br />
Òåîðåìè 9.2.2 – 9.2.4 ïîäàìî áåç äîâåäåííÿ.<br />
9.3. ÂËÀÑÒÈÂÎÑÒ² ÂÈÇÍÀ×ÅÍÎÃÎ<br />
²ÍÒÅÃÐÀËÀ<br />
1. Âåëè÷èíà âèçíà÷åíîãî ³íòåãðàëà íå çàëåæèòü â³ä ïîçíà÷åííÿ<br />
çì³ííî¿ ³íòåãðóâàííÿ:<br />
b b b<br />
( ) = ( ) ... = ( )<br />
∫f x dx ∫f t dt ∫ f y dy .<br />
a a a<br />
2. ßêùî âåðõíÿ ìåæà ³íòåãðóâàííÿ äîð³âíþº íèæí³é, òî<br />
³íòåãðàë äîð³âíþº íóëþ, òîáòî<br />
a<br />
∫ fxdx= ( ) 0.<br />
a<br />
3. ³ä ïåðåñòàâëåííÿ ìåæ ³íòåãðóâàííÿ ³íòåãðàë çì³íþº<br />
çíàê íà ïðîòèëåæíèé, òîáòî<br />
b<br />
∫fxdx<br />
( ) =−∫ fxdx ( ) .<br />
a<br />
a<br />
b<br />
294 295
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