MATEMAATILINE ANALÜÜS II - Tallinna Tehnikaülikool
MATEMAATILINE ANALÜÜS II - Tallinna Tehnikaülikool
MATEMAATILINE ANALÜÜS II - Tallinna Tehnikaülikool
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202 PEATÜKK 3. INTEGRAALARVUTUS<br />
ses suunas. Kasutades Lause 3.10.2 teist osa, saame<br />
∮<br />
Xdx = ∫<br />
Xdx + ∫<br />
Xdx + ∫<br />
Xdx + ∫<br />
Xdx =<br />
Kuna<br />
siis<br />
Γ<br />
⎡<br />
= ⎣<br />
⎡<br />
⎣<br />
[(3.10.6)]<br />
=<br />
= −<br />
AB<br />
BC<br />
CE<br />
AB :<br />
x = x, y = ϕ (x) , dx = dx<br />
−−−−−−→<br />
a ≤ x ≤ b<br />
CE :<br />
x = x, y = ψ (x) , dx = dx<br />
←−−−−−−<br />
a ≤ x ≤ b<br />
=<br />
∫ b<br />
a<br />
∫ b<br />
a<br />
∫ b<br />
a<br />
EA<br />
BC :<br />
x = b, y = y, dx = 0<br />
−−−−−−−−−−−−→<br />
ϕ (b) ≤ y ≤ ψ (b)<br />
EA :<br />
x = a, y = y, dx = 0<br />
←−−−−−−−−−−−−<br />
ϕ (a) ≤ y ≤ ψ (a)<br />
a∫<br />
X (x, ϕ (x)) dx + 0 + X (x, ψ (x)) dx + 0 =<br />
(X (x, ψ (x)) − X (x, ϕ (x))) dx.<br />
∫∫<br />
X y dx dy =<br />
D<br />
∫ b<br />
a<br />
dx<br />
b<br />
ψ(x)<br />
∫<br />
ϕ(x)<br />
X y (x, y) dy =<br />
X (x, y)| ψ(x) b<br />
ϕ(x) dx = ∫<br />
(X (x, ψ (x)) − X (x, ϕ (x))) dx,<br />
∮<br />
Γ<br />
a<br />
∫∫<br />
Xdx = −<br />
D<br />
X y dx dy.<br />
2 ◦ Olgu piirkond D jaotatav y-teljega paralleelsete sirgl~oikudega m x-telje suhtes<br />
normaalseks piirkonnaks D k vastavalt rajajoontega Γ k .<br />
⎤<br />
⎤<br />
⎦<br />
⎦ =<br />
❅■ ✁☛<br />
❅❅<br />
✓✴ ❅❅■<br />
✻❄ ❅✁ ✁✁✁✁✁<br />
✻ ❄<br />
D 3 D<br />
✻<br />
4<br />
✓✴<br />
D5 D m<br />
✤✜ ✜✜✜✜✜✜ ✻❄ ✤✲<br />
✜<br />
❄ D D 2 ❅❘<br />
✻❄<br />
1 ✻❄<br />
✣✘✿<br />
✢ ✣❍❥<br />
✲ ✢
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