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I := x 2 dxdy.
D = {(x, y) ∈ R 2 : 1 ≤ x 2 + y 2 ≤ 2}
D
D γ1 γ2
1 √ 2 Br := {(x, y) ∈ R 2 : x 2 + y 2 < r} D = B2 \ B1
D
x 2
dxdy =
B2
x 2
dxdy −
(P (x, y) dx + Q(x, y) dy) =
γi
Bi
∂Q
∂x
B1
x 2 dxdy.
∂P
− dxdy,
∂y
P : R 2 → R Q : R 2 → R
Bi
P Q
x 2 Q(x, y) = x 3 /3 P (x, y) = 0
I := 1
3
γ2
x 3 dy − 1
3
γ1
x 3 dy.
γi
I := 1
3
=
= 1
16
2π
( √ 2 cos θ) 3 · ( √ 2 cos θ) dθ − 1
3
2π
cos 3 θ · (cos θ) dθ =
0
0
2π 4
eiθ + e−iθ dθ =
0 2
1
2π
(e
16 0
2iθ + e −2iθ + 2) 2 dθ
2π
(e
0
4iθ + e −4iθ + 4 + 2 + 4e 2iθ + 4e −2iθ ) dθ
2π
= 1
8
0
(cos 4θ + 4 cos 2θ + 3) dθ = 3
4 π.
3π/4
R 3
S = {x 2 + y 2 + z 2 = 1, z > 0}
F (x, y, z) =
D = {x 2 + y 2 ≤ 1, z = 0}
γ = {x 2 + y 2 = 1, z = 0}
y
1 + z2 , x5z 100 − y, z + x 2
2π
0
cos 4 θ dθ