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äå<br />
Îòæå,<br />
0≤≤<br />
i n<br />
S<br />
1 1<br />
= lim ρ ( ξ ) ∆ϕ = ρ ( ϕ)<br />
dϕ,<br />
2 2<br />
n<br />
β<br />
2 2<br />
OAB ∑<br />
i i ∫<br />
λ→ 0i= 1<br />
α<br />
λ= max∆ϕ — ðàíã ðîçáèòòÿ ñåãìåíòà [α, β] íà ÷àñòèíí³.<br />
i<br />
Òàêèì ÷èíîì, ïëîùà êðèâîë³í³éíîãî ñåêòîðà (çà óìîâè,<br />
ùî âîíà ³ñíóº) îá÷èñëþºòüñÿ çà ôîðìóëîþ<br />
β<br />
1 2<br />
S = ∫ ρ ( ϕ)<br />
dϕ. (9.7.3)<br />
2 α<br />
Ïðèêëàä 9.7.3. Îá÷èñëèòè ïëîùó ô³ãóðè, ÿêà îáìåæåíà<br />
ρ= 2a<br />
1+ cos ϕ , 0≤ϕ≤2π (ðèñ. 9.19).<br />
êàðä³î¿äîþ ( )<br />
Ðèñ. 9.19<br />
Ð î ç â ’ ÿ ç à í í ÿ. Ô³ãóðà, ÿêà îáìåæåíà êàðä³î¿äîþ (ðèñ.<br />
9.19), íàãàäóº ñåðöå ³ ñèìåòðè÷íà â³äíîñíî îñ³ Îõ. Òîìó ¿¿<br />
ïëîùó ìîæíà îá÷èñëèòè ÿê ïîäâîºíó ïëîùó ¿¿ âåðõíüî¿<br />
÷àñòèíè. Äëÿ íå¿ 0 ≤ϕ≤π, òîìó<br />
1<br />
2<br />
π<br />
= ⋅ ( + ϕ) ϕ= ∫( + ϕ+ ϕ)<br />
ϕ=<br />
2<br />
π<br />
2 2 2<br />
S 2 ∫4a 1 cos d 4a 1 2cos cos d<br />
0 0<br />
π<br />
2 1 cos 2<br />
π<br />
⎛<br />
+ ϕ⎞<br />
2<br />
= 4a ∫⎜1+ 2cosϕ+ ⎟dϕ= 2a ∫( 3+ 4cosϕ+ cos2ϕ)<br />
dϕ=<br />
0⎝<br />
2 ⎠<br />
0<br />
ÂÏÐÀÂÈ<br />
⎛<br />
1 ⎞<br />
= 2 ⎜3ϕ+ 4sinϕ+ sin2ϕ ⎟ = 6π<br />
⎝<br />
2 ⎠<br />
2 π 2<br />
a<br />
0<br />
a .<br />
9.20. Îá÷èñëèòè ïëîùó ô³ãóðè, ÿêà îáìåæåíà òàêèìè ë³í³ÿìè:<br />
1) ïàðàáîëîþ 4y =8x – x 2 ³ ïðÿìîþ 4y = x +6;<br />
2) ïàðàáîëàìè y =4–x 2 ³ y = x 2 –2x;<br />
3) êðèâèìè ρ= 2 3a<br />
cosϕ ³ ρ= 2a<br />
sinϕ.<br />
9.7.2. Îá÷èñëåííÿ äîâæèíè äóã êðèâèõ ë³í³é<br />
1. Äîâæèíà äóãè ó ïðÿìîêóòí³é äåêàðòîâ³é ñèñòåì³<br />
êîîðäèíàò. Íåõàé çàäàíà äóãà À ãðàô³êà ôóíêö³¿ ó = f(x),<br />
ÿêó áóäåìî ââàæàòè íåïåðåðâíî ä³ôåðåíö³éîâíîþ íà cåãìåíò³<br />
[à, b]. Ðîç³á’ºìî ñåãìåíò [à, b] äîâ³ëüíî îáðàíèìè òî÷êàìè<br />
ä³ëåííÿ íà ÷àñòèíí³: à = õ 0 < õ 1
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