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Âèõîäÿ÷è ç ãåîìåòðè÷íîãî çì³ñòó âèçíà÷åíîãî ³íòåãðàëà<br />
íåâàæêî ïîáà÷èòè, ùî ïëîùà ö³º¿ ô³ãóðè äîð³âíþº ð³çíèö³<br />
ïëîù äâîõ êðèâîë³í³éíèõ òðàïåö³é<br />
( ) ( ) ( ( ) ( ))<br />
b b b<br />
SABCD = SaADb − SaBCb<br />
= ∫f x dx− ∫g x dx = ∫ f x −g x dx. (9.7.1)<br />
a a a<br />
Çàóâàæåííÿ. Ôîðìóëà (9.7.1) çáåð³ãàºòüñÿ ³ ó òîìó<br />
âèïàäêó, êîëè ôóíêö³¿ f(õ), g(õ) ïðèéìàþòü áóäü-ÿê³ çíà÷åííÿ,<br />
íå îáîâ’ÿçêîâî íåâ³ä’ºìí³. Àëå ïðè öüîìó ïîâèííà âèêîíóâàòèñÿ<br />
íåð³âí³ñòü f(õ) ≥ g(õ), íàïðèêëàä, ÿê çîáðàæåíî íà<br />
ðèñ. 9.15.<br />
Ðèñ. 9.15<br />
Îá´ðóíòóâàííÿ ôîðìóëè (9.7.1) ó öüîìó âèïàäêó òàêå:<br />
êð³ì ôóíêö³é f(õ), g(õ), ðîçãëÿíåìî ôóíêö³¿ f(õ) +Ñ, g(õ)+Ñ,<br />
äå Ñ âèáèðàºòüñÿ òàê, ùîá íîâ³ ôóíêö³¿ ïðèéìàëè ò³ëüêè<br />
äîäàòí³ çíà÷åííÿ. Äëÿ òàêèõ ôóíêö³é ôîðìóëà (9.7.1) ñïðàâåäëèâà.<br />
Äàë³ òðåáà ñêîðèñòàòèñÿ î÷åâèäíîþ òîòîæí³ñòþ:<br />
(f(õ) +Ñ) –(g(õ) +Ñ) =f(õ) –g(õ).<br />
Ïðèêëàä 9.7.1. Îá÷èñëèòè ïëîùó ô³ãóðè, ÿêó îáìåæåíî<br />
2<br />
ãðàô³êàìè ôóíêö³é y = x+ 2, y = x + 2x<br />
(ðèñ. 9.16).<br />
Ðèñ. 9.16<br />
Ðîçâ’ÿçàííÿ. Ñïî÷àòêó çíàéäåìî ìåæ³ ³íòåãðóâàííÿ.<br />
Äëÿ öüîãî òðåáà çíàéòè àáñöèñè òî÷îê ïåðåòèíó ãðàô³ê³â<br />
2<br />
ôóíêö³é y = x+ 2, y = x + 2x. Îñê³ëüêè â òî÷êàõ ïåðåòèíó îðäèíàòè<br />
îäíàêîâ³, òî äëÿ çíàõîäæåííÿ øóêàíèõ àáñöèñ ìàòèìåìî<br />
ð³âíÿííÿ<br />
2<br />
x + 2x = x+<br />
2<br />
³ çâ³äñè ìàºìî: x 1 = –2, x 2 =1.<br />
Îòæå,<br />
1 1<br />
2 3<br />
2 2<br />
⎛ x x ⎞ 9<br />
S = ∫ ( x+ 2−x − 2x) dx = ∫ ( 2−x− x ) dx = ⎜2x− − ⎟ = = 4,5<br />
−2 −2<br />
2 2 2<br />
.<br />
⎝<br />
⎠ −2<br />
2. Ïëîùà ô³ãóðè, ÿêà îáìåæåíà ë³í³ÿìè, ùî çàäàí³<br />
ïàðàìåòðè÷íî. Íåõàé êðèâîë³í³éíà òðàïåö³ÿ îáìåæåíà êðèâîþ,<br />
ÿêà çàäàíà ïàðàìåòðè÷íî<br />
x =ϕ t , y =ψ t , α≤t ≤β,<br />
() ()<br />
äå ϕ(t), ó = ψ(t) — íåïåðåðâí³ ³ íåïåðåðâíî äèôåðåíö³éîâí³<br />
íà ñåãìåíò³ [α, β] ôóíêö³¿. ßêùî ôóíêö³ÿ ϕ(t) ìîíîòîííà íà<br />
[α, β] ³ ϕ(α) =à, ϕ(β) =b, òî ïëîùà êðèâîë³í³éíî¿ òðàïåö³¿<br />
îá÷èñëþºòüñÿ çà ôîðìóëîþ:<br />
β<br />
() ()<br />
S = ∫ ψ t ϕ′ t dt . (9.7.2)<br />
α<br />
1<br />
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