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4. ßêùî ôóíêö³ÿ ó = f(õ) ³íòåãðîâíà íà ñåãìåíò³ [à, b],<br />
òî ñïðàâåäëèâà ð³âí³ñòü<br />
b c b<br />
( ) = ( ) + ( ) ,<br />
∫f x dx ∫f x dx ∫f x dx<br />
a a c<br />
äå ñ∈(a, b).<br />
Ïðè f(x) ≥ 0 öÿ âëàñòèâ³ñòü äîáðå ³ëþñòðóºòüñÿ ãåîìåòðè÷íî<br />
(ðèñ. 9.6) (S aABb = S I + S II ).<br />
Ðèñ. 9.6<br />
5. Ñòàëèé ìíîæíèê ìîæíà âèíîñèòè çà çíàê âèçíà÷åíîãî<br />
³íòåãðàëà<br />
b<br />
( ) = ∫ ( ) .<br />
∫Cf x dx C f x dx<br />
a<br />
Ä î â å ä å í í ÿ. Çã³äíî ç îçíà÷åííÿì âèçíà÷åíîãî ³íòåãðàëà<br />
ìàºìî:<br />
( ξ ) ∆ = ∫ ( )<br />
lim∑<br />
f x f x dx.<br />
ρ→ 0 i=<br />
1<br />
Ðîçãëÿíåìî òåïåð ôóíêö³þ Cf(x). Òîä³<br />
b<br />
n<br />
i<br />
n<br />
n<br />
( ) = lim∑<br />
( ξ ) ∆ = lim∑<br />
( ξ ) ∆ = ∫ ( )<br />
∫Cf x dx Cf x C f x C f x dx.<br />
a<br />
i i i i<br />
ρ→0 i= 1 ρ→0<br />
i=<br />
1<br />
Ïîð³âíþþ÷è ñàìó ë³âó ³ ñàìó ïðàâó ÷àñòèíè ö³º¿ ð³âíîñò³,<br />
âïåâíþºìîñÿ ó ñïðàâåäëèâîñò³ âëàñòèâîñò³ 5.<br />
i<br />
b<br />
a<br />
b<br />
a<br />
b<br />
a<br />
b<br />
∫<br />
a<br />
6. Cdx = C( b −a).<br />
Äîâåäåííÿ. Äëÿ ôóíêö³¿ f(x) =Ñ ³íòåãðàëüíà ñóìà<br />
äëÿ áóäü-ÿêîãî ðîçáèòòÿ R òàêà:<br />
n<br />
n<br />
i<br />
i= 1 i=<br />
1<br />
( )<br />
∑C∆ x = C∑∆ x = C b−a<br />
.<br />
Òóò ìè ñêîðèñòàëèñÿ òèì, ùî äëÿ áóäü-ÿêîãî ðîçáèòòÿ R<br />
ñóìà ÷àñòèííèõ ñåãìåíò³â çàâæäè äîð³âíþº äîâæèí³ ñåãìåíòà.<br />
Ïåðåéäåìî òåïåð ó ë³â³é ³ ïðàâ³é ÷àñòèíàõ îñòàííüî¿ ð³âíîñò³<br />
äî ãðàíèö³ ïðè ρ→ 0. Îñê³ëüêè ãðàíèöÿ â³ä ñòàëî¿<br />
äîð³âíþº ò³é ñàì³é ñòàë³é, òî çà îçíà÷åííÿì 9.2.1 ä³éñíî<br />
ñïðàâäæóºòüñÿ âëàñòèâ³ñòü 6.<br />
7. Âèçíà÷åíèé ³íòåãðàë â³ä ñóìè (ð³çíèö³) ³íòåãðîâíèõ íà<br />
ñåãìåíò³ [a, b] ôóíêö³é f(x) i ϕ(x) äîð³âíþº ñóì³ (ð³çíèö³)<br />
³íòåãðàë³â â³ä öèõ ôóíêö³é, òîáòî<br />
( ( ) ±ϕ ( )) = ( ) ± ϕ( )<br />
b b b<br />
∫ f x x dx ∫f x dx ∫ x dx.<br />
a a a<br />
Äîâåäåííÿ ïðîâåäåìî äëÿ ñóìè. Ðîç³á’ºìî äîâ³ëüíî<br />
ñåãìåíò [a, b] íà ÷àñòèíí³. Çã³äíî ç óìîâîþ ³íòåãðîâíîñò³<br />
ìàòèìåìî, ùî<br />
n<br />
ρ→ 0 i=<br />
1<br />
i<br />
( ξ ) ∆ = ( )<br />
lim∑ f<br />
i<br />
xi<br />
∫ f x dx ,<br />
n<br />
ρ→ 0i=<br />
1<br />
( ) ( )<br />
lim∑<br />
ϕξi<br />
∆ xi<br />
= ϕx dx .<br />
Äàë³, ò³ ñàì³ ïðîì³æí³ òî÷êè ξ i â³çüìåìî äëÿ íîâî¿ ôóíêö³¿<br />
ψ ( x) = f( x) +ϕ ( x)<br />
, ñêëàäåìî äëÿ íå¿ ³íòåãðàëüíó ñóìó ³<br />
ïåðåéäåìî äî ãðàíèö³. Îñê³ëüêè ôóíêö³¿ f(x) i ϕ(x) ³íòåãðîâí³<br />
íà ñåãìåíò³ [a, b], òî ìàòèìåìî<br />
b<br />
a<br />
b<br />
∫<br />
a<br />
( ) lim∑<br />
( ) lim∑( ( ) ( ))<br />
b n n<br />
∫ ψ xdx= ψ ξ ∆ x= fξ +ϕ ξ ∆ x=<br />
a<br />
n<br />
i i i i i<br />
ρ→0i= 1 ρ→0i=<br />
1<br />
n<br />
( ) lim ( ) ( ) ( )<br />
= lim∑f ξ ∆ x + ∑ϕ ξ ∆ x = ∫f x dx+ ∫ϕ<br />
x dx.<br />
i i i i<br />
ρ→0i= 1 ρ→0i=<br />
1<br />
a a<br />
b<br />
b<br />
296 297
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