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[ñ, b] ñåãìåíòà [a, b] ¿õ ìîæå çîâñ³ì íå áóòè. ßñíî, ùî ïðè<br />
òàêîìó çá³ëüøåíí³ n òî÷í³ñòü íàáëèæåíî¿ ôîðìóëè (9.1.1) º<br />
ïðîáëåìàòè÷íîþ. Îòæå, ãîëîâíèì â óòî÷íåíí³ ôîðìóëè<br />
(9.1.1) º òå, ùîá óñ³ ∆x i (i =1,n) áóëè ñïðÿìîâàí³ äî íóëÿ.<br />
Ðèñ. 9.3<br />
Ç ðèñ. 9.3 âèäíî, ùî ïëîùà åëåìåíòàðíî¿ êðèâîë³í³éíî¿<br />
ñìóãè x i-1 A i-1 A i x i á³ëüøà ïëîù³ ïðÿìîêóòíèêà x i-1 A i-1 B i x i ³ â<br />
ñâîþ ÷åðãó âîíà ìåíøà ïëîù³ ïðÿìîêóòíèêà x i-1 B i-1 A i x i , òîáòî<br />
ìຠì³ñöå ïîäâ³éíà íåð³âí³ñòü<br />
( ) ( )<br />
f xi− 1<br />
∆ xi < Si < f xi ∆ xi. (9.1.2)<br />
²ç ïîäâ³éíî¿ ð³âíîñò³ (9.1.2) âèïëèâຠòàêà:<br />
( −1) ( ( ) ( −1)<br />
)<br />
0 < S −f x ∆ x < f x − f x ∆ x . (9.1.3)<br />
i i i i i i<br />
Îñê³ëüêè çà ïðèïóùåííÿì ôóíêö³ÿ f(x) äèôåðåíö³éîâíà<br />
íà [a, b], òî íà ñåãìåíò³ [x i-1 , x i ] äî íå¿ ìîæíà çàñòîñóâàòè<br />
ôîðìóëó Ëàãðàíæà:<br />
( ) ( ) ( )<br />
i<br />
−<br />
i− 1<br />
= ′<br />
i<br />
∆<br />
i.<br />
f x f x f c x<br />
Âðàõîâóþ÷è öå, ïîäâ³éíà íåð³âí³ñòü (9.1.3) íàáóâຠâèãëÿäó:<br />
( ) ( )( ) 2<br />
1<br />
0 < S −f x ∆ x < f c ∆x<br />
.<br />
′<br />
i i− i i i<br />
Òåïåð ñòຠî÷åâèäíèì, ùî äëÿ òîãî ùîá óòî÷íèòè íàáëèæåíó<br />
ôîðìóëó, òðåáà, ùîá óñ³ âåëè÷èíè ∆x i çìåíøóâàëèñÿ.<br />
Çàóâàæèìî ïðè öüîìó, ùî çá³ëüøåííÿ ê³ëüêîñò³ n ÷àñòèííèõ<br />
cåãìåíò³â íà ñóòòºâå óòî÷íåííÿ ìîæå ³ íå âïëèíóòè<br />
(ðèñ. 9.4). ijéñíî, íà ÷àñòèí³ [a, c] ñåãìåíòà [a, b] òî÷îê<br />
ðîçáèòòÿ R ìîæå áóòè ÿê çàâãîäíî áàãàòî, à íà ÷àñòèí³<br />
Ðèñ. 9.4<br />
ßê öå çä³éñíèòè? Ìîæíà çä³éñíèòè öå òàê: ââåñòè â ðîçãëÿä<br />
âåëè÷èíó ρ= max∆x i (ìàêñèìóì äîâæèíè óñ³õ ÷àñòèííèõ<br />
ñåãìåíò³â) ³ ñïðÿìóâàòè ¿¿ äî íóëÿ. Î÷åâèäíî, ùî ïðè<br />
i=<br />
1, n<br />
öüîìó óñ³ ∆x i ( i = 1, n) òåæ ïðÿìóþòü äî íóëÿ.<br />
Òåïåð ç’ÿñóºìî, ùî æ ìè áóäåìî ðîçóì³òè ï³ä ïëîùåþ<br />
êðèâîë³í³éíî¿ òðàïåö³¿.<br />
Îçíà÷åííÿ 9.1.1. Çà ïëîùó êðèâîë³í³éíî¿ òðàïåö³¿ áóäåìî<br />
ââàæàòè ãðàíèöþ ïîñë³äîâíîñòåé ïëîù ñõ³ä÷àñòèõ ô³ãóð<br />
(ðèñ. 9.2), ÿêùî ìàêñèìàëüíà äîâæèíà ÷àñòèííèõ ñåãìåíò³â<br />
ïðÿìóº äî íóëÿ, òîáòî<br />
ρ→ 0 i=<br />
1<br />
( )<br />
S= lim∑ n f ξ . i<br />
∆xi (9.1.4)<br />
9.1.2. Çàäà÷à ïðî ðîáîòó çì³ííî¿ ñèëè<br />
Íåõàé âçäîâæ îñ³ Îõ 䳺 ñèëà Ð(õ), íàïðÿì ÿêî¿ ñòàëèé<br />
³ çá³ãàºòüñÿ ç íàïðÿìîì îñ³ Îõ. Êð³ì òîãî, ñèëà íåïåðåðâíî<br />
çì³íþºòüñÿ çà âåëè÷èíîþ. Íåõàé ï³ä 䳺þ ñèëè Ð(õ) ìàòå-<br />
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