ЛЕКЦІЇ ² ВПРАВИ

Îñê³ëüêè ôóíêö³ÿ ó = f(x) íåïåðåðâíî ä³ôåðåíö³éîâíà íà
cåãìåíò³ [à, b], òî çã³äíî ç ôîðìóëîþ Ëàãðàíæà íà êîæíîìó
÷àñòèííîìó ñåãìåíò³ [x i-1 , x i ] ìàòèìåìî
Òîä³
( ) ( 1 ) ( )
∆ y = f x − f x = f ξ ⋅∆ x .
′
i i i− i i
n
n
( ′( )) ( ′( ))
2 2
i
1
i i, n i
1
i i
i= 1 i=
1
∆ S = + f ξ ⋅∆ x P = ∑∆ S = ∑ + f ξ ⋅∆x
.
Îñòàííÿ ñóìà ÿâëÿº ñîáîþ ³íòåãðàëüíó ñóìó äëÿ ôóíêö³¿
( ′( )) 2
1 + f x . Îñê³ëüêè f′(x) íåïåðåðâíà, òî ôóíêö³ÿ
( ′( )) 2
1 + f x òåæ íåïåðåðâíà, ³ òîä³ ³ñíóº â³äïîâ³äíà ãðàíèöÿ
(äèâ. ï. 9.2):
1 3
4 4
3 3 2 2
9 ⎛ 9 ⎞ 4 2⎛ 9 ⎞ 56
l = ∫ 1+ xdx = ∫ ⎜1+ x⎟ dx = ⋅ ⎜1+ x ⎟ = .
0 4 0 ⎝ 4 ⎠ 9 3⎝ 4 ⎠ 27
2. Äîâæèíà äóãè ó âèïàäêó ¿¿ ïàðàìåòðè÷íîãî çàäàííÿ.
ßêùî êðèâà À çàäàíà ïàðàìåòðè÷íî: õ = ϕ(t), ó = ψ(t),
α≤t ≤β, äå ϕ, ψ — íåïåðåðâíî äèôåðåíö³éîâí³ íà ñåãìåíò³
[α, β] ôóíêö³¿, òî
β
α
( ′()) ( ′())
2 2
l = ∫ ϕ t + ψ t dt . (9.7.5)
Ïðèêëàä 9.7.5. Îá÷èñëèòè äîâæèíó îäí³º¿ àðêè öèêëî¿-
äè (ðèñ. 9.21), ÿêà çàäàíà ïàðàìåòðè÷íèìè ð³âíÿííÿìè:
õ = à(t – sint), ó = à(1– ñîst) (0 ≤ t ≤ 2π).
4
3
0
n
λ→ 0 i=
1
b
( ′( )) ( ′
i
i ( ))
2 2
l = lim∑ 1+ f ξ ⋅∆ x = ∫ 1+
f x dx.
Îòæå, îòðèìàëè ôîðìóëó:
a
b
( ′( )) 2
l = ∫ 1 + f x dx. (9.7.4)
a
Ïðèêëàä 9.7.4. Îá÷èñëèòè äîâæèíó äóãè íàï³âêóá³÷íî¿
3
ïàðàáîëè y = x 2
4 , x ∈ [0, ] (ðèñ. 9.20).
3
Ðèñ. 9.21
Ð î ç â ’ ÿ ç à í í ÿ. Öèêëî¿äà — öå ë³í³ÿ, ÿêó îïèñóº òî÷êà
íà êîë³ ðàä³óñà à, ÿêå êîòèòüñÿ âçäîâæ ïðÿìî¿ ë³í³¿. Ó
ÿêîñò³ ïàðàìåòðà t âèñòóïຠêóò ïîâîðîòó êîëà.
Çà ôîðìóëîþ (9.7.5) ìàºìî:
2 2
2π
2π
⎛ ′ ⎞ ⎛ ′ ⎞
2 2 2 2
l = ∫ ⎜( a( t− sin t)
) ⎟ + ⎜( a( 1− cost)
) ⎟ = ∫ a( 1− cost)
+ a sin t dt =
0 ⎝ ⎠ ⎝ ⎠ 0
Ðèñ. 9.20
Ð î ç â ’ ÿ ç à í í ÿ. Çàñòîñóºìî ôîðìóëó (9.7.4). Òîä³ ìàòèìåìî
2π
2π
2π
t
t
= a ∫ 2− 2cos t dt = 2a ∫ sin dt = − 4acos 0
= 8a .
0 0 2 2
3. Äîâæèíà äóãè â ïîëÿðí³é ñèñòåì³ êîîðäèíàò. ßêùî
êðèâà çàäàíà ó ïîëÿðí³é ñèñòåì³ êîîðäèíàò: ρ = ρ(ϕ),
324 325