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Ïðèêëàä 9.7.2. Îá÷èñëèòè ïëîùó ô³ãóðè, ÿêà îáìåæåíà<br />
åë³ïñîì õ = acos t, y = b sin t, 0 ≤ t ≤ 2π (äèâ. ï. 4.4.1).<br />
Ðîçâÿçàííÿ. Î÷åâèäíî, ùî øóêàíà ïëîùà ìîæå áóòè<br />
çíàéäåíà ìíîæåííÿì íà 4 ïëîù³ ¿¿ ÷àñòèíè, ùî ðîçòàøîâàíà<br />
ó 1-ìó êâàäðàíò³. Äëÿ ö³º¿ ÷àñòèíè:<br />
π<br />
⎛π⎞<br />
0 ≤t ≤ , ϕ ( t) = acos t, ϕ (0) = a , ϕ ⎜ ⎟=<br />
0.<br />
2 ⎝2⎠<br />
Òîìó çã³äíî ç ôîðìóëîþ (9.7.2) ìàòèìåìî:<br />
π<br />
0 0 2<br />
′<br />
2<br />
S = 4 ∫ bsin t ( a cos t) dt =− 4ab ∫ sin tdt = 2ab ∫ ( 1− cos 2t)<br />
dt =<br />
π<br />
π<br />
2 2<br />
π<br />
2<br />
⎛ 1 ⎞<br />
= 2ab⎜t − sin2t ⎟ = πab<br />
.<br />
⎝ 2 ⎠<br />
Îòæå, ïëîùà ô³ãóðè, ÿêà îáìåæåíà åë³ïñîì, äîð³âíþº<br />
πab. Öåé ðåçóëüòàò ïîâí³ñòþ çá³ãàºòüñÿ ç ðåçóëüòàòîì<br />
îá÷èñëåííÿ ïëîù³ ô³ãóðè, îáìåæåíî¿ åë³ïñîì, ³íøèì ñïîñîáîì<br />
(äèâ. ïðèêë. 9.5.1).<br />
3. Ïëîùà ô³ãóðè ó ïîëÿðí³é ñèñòåì³ êîîðäèíàò. Ðîçãëÿíåìî<br />
ô³ãóðó ÎÀÂ, ÿêà îáìåæåíà êðèâîþ, çàäàíîþ ó ïîëÿðí³é<br />
ñèñòåì³ êîîðäèíàò ρ=ρ( ϕ ) ³ ïðîìåíÿìè ϕ=α,<br />
ϕ=β<br />
(ðèñ. 9.17).<br />
0<br />
0<br />
Òàêà ô³ãóðà íàçèâàºòüñÿ êðèâîë³í³éíèì ñåêòîðîì. Îá÷èñëèìî<br />
éîãî ïëîùó. Ðîç³á’ºìî ñåãìåíò [α, β] äîâ³ëüíî îáðàíèìè<br />
òî÷êàìè<br />
α=ϕ<br />
0<br />
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