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p = 2
|(x, y)| −p dx dy =
R 2 \B
2π +∞
p > 2 p = 2
∞ 1
2π
1 ρ = 2π[log ρ]∞1 = +∞,
R2 \ B p > 2
0
1
1
2π
ρ dρ dθ =
ρp (2 − p) [ρ2−p ] ρ=∞
ρ=1
x = ρ sin φ cos θ y = ρ sin φ sin θ z = ρ cos φ
ρ2 sin φ p = 3
π 2π 1
0 0 0
ρ 2−p sin φ dρ dθ dφ = 4π
3 − p
p < 3 p = 3
1
π/2
4π
0
1
= +∞
ρ
B(0, 1) ⊆ R3 p < 3
p = 3
R 3 \B(0,1)
p > 3 p = 3
0
sin φ dφ · [ρ 3−p ] ρ=1 4π
ρ=0 =
3 − p [ρ3−p ] ρ=1
ρ=0
|(x, y, z)| −p dx dy dz = 4π
3 − p [ρ3−p ] ρ=+∞
ρ=1
R3 |(x, y, z)|
\B(0,1)
−p dx dy dz = 4π[log ρ] ∞ 1 = +∞.
R 3 \ B(0, 1) p > 3
I := dx dy
D ρ = 1 + cos θ θ ∈ [0, π]
x
D = {(s cos θ, s sin θ) : θ ∈ [0, π], s ∈ [0, 1 + cos θ]}
π 1+cos θ
I :=
ρ dρ dθ =
0 0
1
(1 + cos θ)
2 0
2 dθ
= 1
π
dθ +
2 0
1
π
2 cos θ dθ +
2 0
1
π
cos
2 0
2 θ dθ
= π π 3
+ =
2 4 4 π.
D := {(x, y) ∈ R 2 : (x + 1) 2 + y 2 ≥ 1, (x − 1) 2 + y 2 ≥ 1, x 2 + y 2 ≤ 2}.
f(x, y) =
I =
D
D
π
|y|
x 2 + y 2 ,
f(x, y) dx dy.