Integral triplo
Ointegral triploda fun¸c˜aof sobreB = [a,b]×[c,d]×[r,s] ´e
Z Z Z
B
f(x,y,z)dV = lim
l,m,n→∞
l
X
i=1 m
X
j=1 n
X
k=1
f(xijk∗,yijk∗,zijk∗)∆x∆y∆z
O integral triplo n˜ao representa um volume (a n˜ao ser quef(x,y,z) = 1)
Teorema de Fubini
Sob certas condi¸c˜oes (por exemplo, se f for cont´ınua em B= [a,b]×[c,d]×[r,s]) tem-se:
Z Z Z
B
f(x,y,z)dV = Z Z
R
Z s r
f(x,y,z)dz dxdy (ondeR = [a,b]×[c,d])
= Z b
a
Z d c
Z s r
f(x,y,z)dz dy
dx
= Z b
a
Z s r
Z d c
f(x,y,z)dy dz
dx
= Z d
c
Z s r
Z b a
f(x,y,z)dx dz
dy
= . . .
Sef for cont´ınua emB= [a,b]×[c,d]×[r,s]) tem-se:
Z Z Z
B
f(x,y,z)dV = Z b
a
Z d
c
Z s
r
f(x,y,z)dz dy
dx
= Z d
c
Z s
r
Z b
a
f(x,y,z)dx dz
dy
= . . .
Exerc´ıcio:
Integral triplo sobre uma regi˜ao do tipo 1
SeE =
(x,y,z)∈R3: (x,y)∈D e u1(x,y)≤z ≤u2(x,y) ,
Z Z Z
E
f(x,y,z)dV = Z Z
D
Z u2(x,y) u1(x,y)
f(x,y,z)dz dA
Integral triplo sobre uma regi˜ao do tipo 3
Z Z Z
E
f(x,y,z)dV = Z Z
D
Z u2(x,z) u1(x,z)
f(x,y,z)dy dxdz Exerc´ıcio:
SeE =
(x,y,z)∈R3: (x,y)∈D e u1(x,y)≤z ≤u2(x,y) ,
Z Z Z
E
f(x,y,z)dV = Z Z
D
Z u2(x,y)
u1(x,y)
f(x,y,z)dz dA
Exerc´ıcio:
Coordenadas cil´ındricas
x=rcosθ y =rsinθ z =z
Z Z Z
E
f(x,y,z)dx dy dz = Z Z Z
E∗
f rcosθ,rsinθ,z
r dz dr dθ
ondeE∗=
(r, θ,z) : (rcosθ,rsinθ,z)∈E
Coordenadas cil´ındricas
x =rcosθ y =rsinθ z =z
Z Z Z
E
f(x,y,z)dx dy dz = Z Z Z
E∗
f rcosθ,rsinθ,z
r dz dr dθ ondeE∗=
(r, θ,z) : (rcosθ,rsinθ,z)∈E Exerc´ıcio:
Coordenadas esf´ericas
x=ρsinφcosθ y =ρsinφsinθ z =ρcosφ
Z Z Z
E
f(x,y,z)dxdydz = Z Z Z
E∗
f ρsinφcosθ, ρsinφsinθ, ρcosφ
ρ2sinφdρdθdφ
ondeE∗=
(ρ, θ, φ) : (ρsinφcosθ, ρsinφsinθ, ρcosφ)∈E
Z Z Z
E
f(x,y,z)dxdydz = Z Z Z
E∗
f ρsinφcosθ, ρsinφsinθ, ρcosφ
ρ2sinφdρdθdφ
ondeE∗=
(ρ, θ, φ) : (ρsinφcosθ, ρsinφsinθ, ρcosφ)∈E
Exerc´ıcio:
Coordenadas cil´ındricas: r dz dr dθ
x =rcosθ y =rsinθ z =z
Coordenadas esf´ericas: ρ2sinφdρdθdφ
x=ρsinφcosθ y=ρsinφsinθ z =ρcosφ Exerc´ıcio:
