Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
α≤ϕ≤β, äå ρ(ϕ) — íåïåðåðâíà äèôåðåíö³éîâíà íà [α, β]<br />
ôóíêö³ÿ, òî ìîæíà äîâåñòè, ùî<br />
l<br />
( ) ( ′( )) 2<br />
β<br />
2<br />
= ρ ϕ + ρ ϕ dϕ<br />
α<br />
∫ . (9.7.6)<br />
Ïðèêëàä 9.7.6. Îá÷èñëèòè äîâæèíó äóãè ïåðøîãî âèòêà<br />
ñï³ðàë³ Àðõ³ìåäà (ðèñ. 9.22) ρ= aϕ, 0≤ϕ≤2π.<br />
ÂÏÐÀÂÈ<br />
9.21. Îá÷èñëèòè äîâæèíó äóãè:<br />
1) ï³âêóá³÷íî¿ ïàðàáîëè y 2 =(x –1) 3 ì³æ òî÷êàìè A(2, –1)<br />
³ B(5, –8).<br />
3 ϕ<br />
2) êðèâî¿ ρ= a cos .<br />
3<br />
9.8. ÎÁ×ÈÑËÅÍÍß ÎÁ’ªÌ²Â Ò²Ë<br />
9.8.1. Îá÷èñëåííÿ îá’ºì³â ò³ë çà â³äîìèìè ïëîùàìè<br />
ïåðåð³ç³â<br />
Ðîçãëÿíåìî äåÿêå ò³ëî Ò (ðèñ. 9.23). Ïîçíà÷èìî ÷åðåç<br />
S(õ) ïëîùó ïåðåð³çó öüîãî ò³ëà ïëîùèíîþ, ùî ïðîõîäèòü<br />
ïåðïåíäèêóëÿðíî äåÿêî¿ îñ³ ÷åðåç òî÷êó ç êîîðäèíàòîþ õ<br />
íà ö³é îñ³ (a ≤ x ≤ b).<br />
Ðèñ. 9.22<br />
Ð î ç â ’ ÿ ç à í í ÿ. Çàñòîñóºìî ôîðìóëó (9.7.6). Ó ðåçóëüòàò³<br />
îòðèìàºìî<br />
⎡<br />
2<br />
u = ϕ + 1, dv = dϕ⎤<br />
2π<br />
2π<br />
⎢<br />
⎥<br />
2 2 2 2<br />
l = ∫ a ϕ + a dϕ = a∫ 1 +ϕ dϕ = ⎢ ϕ<br />
⎥ =<br />
0 0<br />
du = dϕ , v = ϕ<br />
2<br />
⎢⎣<br />
ϕ + 1 ⎥⎦<br />
( )<br />
2π 2π 2 2 2<br />
2 ϕ<br />
π<br />
2 ϕ + 1−1<br />
= a ϕ + 1 ϕ −a⋅ ∫ dϕ= 2aπ 4π + 1−a∫<br />
dϕ=<br />
0 0<br />
2<br />
0<br />
2<br />
ϕ + 1 ϕ + 1<br />
( )<br />
2π<br />
2 dϕ<br />
2 2<br />
= 2aπ 4π + 1− l+ a∫<br />
= 4aπ 4π + 1− l+ aln 2π+ 4π + 1<br />
0<br />
2<br />
.<br />
ϕ + 1<br />
Ñàìà ë³âà ÷àñòèíà ³ ñàìà ïðàâà ÷àñòèíà ëàíöþæêà ð³âíîñòåé<br />
ÿâëÿþòü ñîáîþ ð³âíÿííÿ â³äíîñíî øóêàíî¿ äîâæèíè l.<br />
Ðîçâ’ÿçóþ÷è éîãî, îñòàòî÷íî ìàòèìåìî<br />
( )<br />
⎡<br />
1<br />
= ⎢π π + + π+ π +<br />
⎣<br />
2<br />
2 2<br />
l a 4 1 ln 2 4 1 .<br />
⎤<br />
⎥<br />
⎦<br />
Ðèñ. 9.23<br />
Çä³éñíèìî äîâ³ëüíå ðîçáèòòÿ R ñåãìåíòà [a, b] íà ÷àñòèíí³<br />
ñåãìåíòè òî÷êàìè<br />
R: a = x 0 < x 1 < ...< x n–1 < x n = b<br />
³ ïðîâåäåìî ÷åðåç ö³ òî÷êè ïëîùèíè, ïåðïåíäèêóëÿðí³ ñåãìåíòó<br />
[a, b]. Íà êîæíîìó ÷àñòèííîìó ñåãìåíò³ [x i–1 , x i ] âèáåðåìî<br />
äîâ³ëüíó òî÷êó ξ i . Ïëîùèíè ðîçáèâàþòü ò³ëî Ò íà<br />
øàðè, â ÿê³ âïèøåìî åëåìåíòàðí³ öèë³íäðè Ò i . Ïëîùà îñíîâè<br />
öèë³íäðà Ò i äîð³âíþº S(ξ i ), à âèñîòà ∆õ i = õ i – x i-1 . Îá’ºì<br />
âñ³õ òàêèõ öèë³íäð³â îá÷èñëþºòüñÿ çà ôîðìóëîþ<br />
n n<br />
n<br />
=<br />
i<br />
= ( ξi)<br />
∆<br />
i<br />
i= 1 i=<br />
1<br />
V ∑T ∑ S x .<br />
326 327