Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
Ð î ç â ’ ÿ ç à í í ÿ. Çà ôîðìóëîþ (9.9.1) ìàºìî<br />
ÂÏÐÀÂÈ<br />
β<br />
( ) ( ) 2<br />
( ( )) 2<br />
S = 2π∫<br />
a 1− cost asint + a 1− cost dt =<br />
α<br />
t 64<br />
= πa − t dt = π a dt = πa .<br />
2π<br />
3<br />
2π<br />
2 2<br />
2 3 2<br />
2 2 ∫ ( 1 cos ) 8 ∫ sin<br />
0 0 2 3<br />
9.23. Çíàéòè ïëîùó ïîâåðõí³, óòâîðåíî¿ îáåðòàííÿì íàâêîëî<br />
â³ñ³ Ox:<br />
1) äóãè êóá³÷íî¿ ïàðàáîëè y = x 3 , ÿêà ì³ñòèòüñÿ ì³æ ïðÿìèìè<br />
2<br />
x =− ³<br />
3<br />
2<br />
x = ;<br />
3<br />
2 2<br />
x y<br />
2) åë³ïñà + = 1<br />
2 2 , a > b;<br />
a b<br />
3) îêðóæíîñò³ x = acos t, y = asin t;<br />
4) äóãè ïàðàáîëè y 2 =2x ì³æ òî÷êàìè ïåðåòèíó ç ïðÿìîþ<br />
2x =3.<br />
Ïðèêëàä 9.10.1. Îá÷èñëèòè ðîáîòó, ÿêó òðåáà çàòðàòèòè,<br />
ùîá ò³ëî ìàñè m ï³äíÿòè ç ïîâåðõí³ Çåìë³ âåðòèêàëüíî<br />
âãîðó íà âèñîòó h, ÿêùî ðàä³óñ Çåìë³ äîð³âíþº R.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Çã³äíî ç çàêîíîì Íüþòîíà ñèëà çåìíîãî<br />
òÿæ³ííÿ äîð³âíþº:<br />
mM<br />
F( x) =γ<br />
2 ,<br />
x<br />
äå Ì — ìàñà Çåìë³, γ — ãðàâ³òàö³éíà ñòàëà, õ — â³äñòàíü<br />
â³ä öåíòðà ò³ëà äî öåíòðà Çåìë³ (ðèñ. 9.26). Î÷åâèäíî, ùî<br />
R ≤ õ ≤ R + h. ßêùî õ = R, (ò³ëî çíàõîäèòüñÿ íà ïîâåðõí³<br />
Çåìë³), òî (R) =Ð = mg — âàãà ò³ëà, òîáòî<br />
mM<br />
F( R) =γ = P ,<br />
2<br />
R<br />
³ çâ³äñè çà ôîðìóëîþ (9.10.1) ìàºìî:<br />
R+ h<br />
R+<br />
h 2<br />
pR PRh<br />
A= ∫ F( x)<br />
dx = ∫ dx=<br />
2<br />
R<br />
R x R+ h<br />
.<br />
9.10. Ô²ÇÈ×Ͳ ÇÀÑÒÎÑÓÂÀÍÍß<br />
ÂÈÇÍÀ×ÅÍÎÃÎ ²ÍÒÅÃÐÀËÀ<br />
Çàñòîñóâàíü âèçíà÷åíèõ ³íòåãðàë³â ó ô³çèö³ äóæå áàãàòî.<br />
³äçíà÷èìî äåÿê³ ç íèõ.<br />
9.10.1. Ðîáîòà çì³ííî¿ ñèëè<br />
ßê áóëî âæå ñêàçàíî â ï. 9.1.2, îá÷èñëåííÿ ðîáîòè çì³ííî¿<br />
ñèëè ïðèâîäèòü äî ïîíÿòòÿ âèçíà÷åíîãî ³íòåãðàëà. Íàãàäàºìî,<br />
ùî, ÿêùî çì³ííà ñèëà (x) 䳺 âçäîâæ íàïðÿìó<br />
ðóõó íà ñåãìåíò³ [a, b], òî ðîáîòà À îá÷èñëþºòüñÿ çà ôîðìóëîþ<br />
b<br />
a<br />
( )<br />
A= ∫ F x dx . (9.10.1)<br />
Ó öüîìó ïóíêò³ ðîçãëÿíåìî êîíêðåòíå çàñòîñóâàííÿ âèçíà÷åíîãî<br />
³íòåãðàëà.<br />
Ðèñ. 9.26<br />
332 333