MATEMAATILINE ANALÜÜS II - Tallinna Tehnikaülikool
MATEMAATILINE ANALÜÜS II - Tallinna Tehnikaülikool
MATEMAATILINE ANALÜÜS II - Tallinna Tehnikaülikool
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3.15. PINDINTEGRAALIDE RAKENDUSED 221<br />
siis valemite (3.15.2–3.15.6) ja (3.13.2) abil saame<br />
∫∫<br />
m Σ =<br />
pr xyΣ<br />
= √ ∫∫<br />
2<br />
pr xyΣ<br />
2π ∫ 1<br />
= √ ∫<br />
2 dϕ<br />
0<br />
x c = 4 ∫∫<br />
5π √ 2<br />
= 4 ∫∫<br />
5π √ 2<br />
= 4<br />
5π<br />
= 4<br />
5π<br />
∫<br />
2π<br />
0<br />
∫<br />
2π<br />
0<br />
y c = 4<br />
5π √ 2<br />
dϕ<br />
(<br />
x 2 + 2y 2 + ( x 2 + y 2)) √ 2dxdy =<br />
0<br />
(<br />
2x 2 + 3y 2) dxdy =<br />
pr xyΣ<br />
pr xyΣ<br />
∫ 1<br />
0<br />
ρ 3 ( 2 cos 2 ϕ + 3 sin 2 ϕ ) dρ = 5π√ 2<br />
,<br />
4<br />
x ( x 2 + 2y 2 + ( x 2 + y 2)) √ 2dxdy =<br />
x ( 2x 2 + 3y 2) √ 2dxdy =<br />
ρ 4 cos ϕ ( 2 cos 2 ϕ + 3 sin 2 ϕ ) dρ =<br />
(<br />
2 cos 2 ϕ + 3 sin 2 ϕ ) cos ϕdϕ<br />
∫∫<br />
pr xyΣ<br />
∫ 1<br />
0<br />
ρ 4 dρ = 0,<br />
y ( x 2 + 2y 2 + ( x 2 + y 2)) √ 2dxdy =<br />
= 4 ∫2π<br />
(<br />
2 cos 2 ϕ + 3 sin 2 ϕ ) ∫ 1<br />
sin ϕdϕ ρ 4 dρ = 0,<br />
5π<br />
0<br />
0<br />
z c = 4 ∫∫<br />
5π √ √<br />
x2 + y ( 2 x 2 + 2y 2 + ( x 2 + y 2)) √ 2dxdy =<br />
2 pr xyΣ<br />
= 4 ∫∫<br />
5π √ √<br />
x2 + y ( 2 2x 2 + 3y 2) √ 2dxdy =<br />
2<br />
pr xyΣ<br />
= 4 ∫2π<br />
(<br />
2 cos 2 ϕ + 3 sin 2 ϕ ) ∫ 1<br />
dϕ ρ 4 dρ = 4 5π<br />
5 ,<br />
0<br />
0<br />
∫∫<br />
(<br />
I x = y 2 + ( x 2 + y 2)) ( x 2 + 2y 2 + ( x 2 + y 2)) √ 2dxdy =<br />
∫∫<br />
=<br />
pr xyΣ<br />
pr xyΣ<br />
0<br />
(<br />
x 2 + 2y 2) ( 2x 2 + 3y 2) √ 2dxdy =<br />
= √ ∫2π<br />
(<br />
2 cos 2 ϕ + 2 sin 2 ϕ ) ( 2 cos 2 ϕ + 3 sin 2 ϕ ) dϕ<br />
∫ 1<br />
0<br />
ρ 5 dρ = 31π√ 2<br />
,<br />
24
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