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Ãðàíèöÿ ö³º¿ ñóìè ïðè<br />
λ= max ∆x<br />
→ 0 (ÿêùî âîíà ³ñíóº)<br />
1≤≤<br />
i n<br />
íàçèâàºòüñÿ îá’ºìîì äàíîãî ò³ëà. Î÷åâèäíî, ùî V n — ³íòåãðàëüíà<br />
ñóìà äëÿ ôóíêö³¿ S(õ) íà ñåãìåíò³ [à, b]. Îòæå, çà<br />
îçíà÷åííÿì âèçíà÷åíîãî ³íòåãðàëà îá’ºì ò³ëà Ò òàêèé:<br />
b<br />
V = ∫ S( x)<br />
dx . (9.8.1)<br />
a<br />
Çàóâàæåííÿ. ßêùî íàì â³äîì³ ïëîù³ S(y) ïåðåð³ç³â,<br />
ùî ïðîõîäÿòü ïåðïåíäèêóëÿðíî îñ³ Îy ÷åðåç òî÷êó ç êîîðäèíàòîþ<br />
ó íà ö³é îñ³ (ñ ≤ ó ≤ d), òî îá’ºì ò³ëà Ò (çà óìîâè,<br />
ùî â³í ³ñíóº) îá÷èñëþºòüñÿ çà òàêîþ ôîðìóëîþ:<br />
d<br />
( )<br />
i<br />
V = ∫ S y dy . (9.8.2)<br />
c<br />
Ïðèêëàä 9.8.1. Çíàéòè îá’ºì åë³ïñî¿äà (ðèñ. 9.24)<br />
2 2 2<br />
x y z<br />
+ + = 1.<br />
2 2 2<br />
a b c<br />
àáî<br />
2 2 2<br />
x z y<br />
+ = 1− 2 2 2 ,<br />
a c b<br />
x<br />
a<br />
z<br />
+ 1<br />
c<br />
= .<br />
2 2<br />
2 2<br />
1 1<br />
2 2<br />
y<br />
y<br />
ϳâîñ³ öüîãî åë³ïñà òàê³: a = 1<br />
a 1 − , c<br />
2 1<br />
c 1<br />
2<br />
b<br />
= − b<br />
. Éîãî<br />
ïëîùà (äèâ. ïðèêë. 9.5.1, 9.7.2) äîð³âíþº<br />
2<br />
⎛<br />
Sy () ab<br />
1 1<br />
ac 1 y ⎞<br />
=π =π ⎜ −<br />
2 ⎟<br />
⎝ b ⎠ .<br />
Òîä³ çà ôîðìóëîþ 9.8.2 îòðèìàºìî<br />
b<br />
2 2 3 b<br />
⎛ y ⎞ b⎛ y ⎞ ⎛ y ⎞ 4<br />
V =πac∫<br />
⎜1− dy 2 ac 1 dy 2 ac y abc<br />
2 ⎟ = π ∫⎜ −<br />
2 ⎟ = π ⎜ −<br />
2 ⎟ = π<br />
−b<br />
⎝ b ⎠ 0⎝ b ⎠ ⎝ 3b<br />
⎠ 0 3<br />
.<br />
Çîêðåìà, ÿêùî ïîêëàñòè a = b = c = R, òî îòðèìàºìî â³äîìó<br />
ôîðìóëó îá’ºìó êóë³<br />
V<br />
4<br />
3<br />
3<br />
= π R .<br />
9.8.2. Îá’ºì ò³ëà îáåðòàííÿ<br />
Ðîçãëÿíåìî ò³ëî, ÿêå óòâîðåíî îáåðòàííÿì ãðàô³êà ôóíêö³¿<br />
ó = f(x) íàâêîëî cåãìåíòà [à, b] îñ³ Îõ (ðèñ. 9.25). Òîä³<br />
ïëîùà ïåðåð³çó S(x) =πf 2 (õ), ³ çã³äíî ç ôîðìóëîþ (9.8.1)<br />
ìàòèìåìî:<br />
Ðèñ. 9.24<br />
Ð î ç â ’ ÿ ç à í í ÿ. Ó ïåðåð³ç³ åë³ïñî¿äà ïëîùèíîþ, ïàðàëåëüíîþ<br />
ïëîùèí³ Îxyz íà â³äñòàí³ ó â³ä íå¿, óòâîðþºòüñÿ<br />
åë³ïñ<br />
b<br />
2<br />
V f ( x)<br />
dx<br />
a<br />
=π∫ . (9.8.3)<br />
Ïðèêëàä 9.8.2. Çíàéòè îá’ºì ò³ëà, óòâîðåíîãî îáåðòàííÿì<br />
ãðàô³êà ôóíêö³¿ y = sin x íàâêîëî â³äð³çêà [0, π] îñ³<br />
Îõ.<br />
Ð î ç â ’ ÿ ç à í í ÿ. Çà ôîðìóëîþ (9.8.3) ìàºìî:<br />
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