f: R ⊆ ℝ2 → ℝ f
R = {(x, y) ∈ ℝ2 │a ≤ x ≤ b, a ≤ y ≤ b} n lim ∑m i=1 ∑j=1 f(xij∗ , yij∗ )∆A
m,n→∞
m
n
∬ f(x, y)dA = lim ∑ ∑ f(xij∗ , yij∗ )∆A m,n→∞
R n ∑m i=1 ∑j=1 f(xij∗ , yij∗ )∆A
i=1 j=1
𝑧 𝑅ij
𝑖𝑗 f(xij∗ , yij∗ )
(xij∗ , yij∗ ) ∈ 𝑅ij 𝑅i 𝑅
𝑧 = 𝑓(x, y) 𝑅
f: R ⊆ ℝ2 → ℝ y ≤ d} 𝑘 ∈ ℝ
g: R ⊆ ℝ2 → ℝ
R = {(x, y) ∈ ℝ2 │a ≤ x ≤ b, c ≤
f(x, y) ≥ 0 ∬R f(x, y)dA = V
V
𝑧 = f(x, y) ∬R[f(x, y) + g(x, y)dA] = ∬R f(x, y)dA + ∬R g(x, y) dA ∬R kf(x, y)dA = k ∬R f(x, y)dA f(x, y) ≥ g(x, y)
∬R f(x, y)dA ≥ ∬R g(x, y)dA
R = R1 ∪ R 2 ∪ R 3 … . R m f Ri m ∬R f(x, y)dA = ∬R f(x, y)dA + ∬R f(x, y)dA + ⋯ + ∬R f(x, y)dA = ∑i=1 ∬R f(x, y)dA i = 1, … m 1
2
m
i
f: R = {(x, y) ∈ ℝ2 │a ≤ x ≤ b, c ≤ y ≤ d} → ℝ 𝑧 = f(x, y) 𝑥 = 𝑥0 z = f(x0 , y)
xy f(x, y) y
x d
A(x0 ) = âˆŤ f(x0, y)dy c
x
y
b
A(y0 ) = âˆŤ f(x, y0 )dy a
� = f(x, y)
b d
V = âˆŤ[âˆŤ f(x, y)dy dx] a
c
z = f(x, y) A(x) d âˆŤc f(x0,
y)dy
b
A(y) = âˆŤa f(x, y0 )dy
z = f(x, y) = seny + cosx � = [0, π] × [0, π] π
Ď€
âˆŤ0 [âˆŤ0 seny + cosx dx]dy
x π
∫ (seny + cosx) dx = xseny + senx│π0 = πseny 0
y
π
∫ πsenydy = −πcosy│π0 = 2π 0
π
π
∫0 [∫0 senx + cosy dx]dy = 2π
z = f(x, y) = seny + cosx z
xy
R
f: R ⊆ ℝ2 → ℝ
R = [a, b]X[c, d]
b d
d
b
∬ f(x, y)dA = ∫ ∫ f(x, y)dydx = ∫ ∫ f(x, y)dxdy . R
a c
c
a
𝑧 = 2𝑥 2 + 4𝑦 𝑅 = [0,4]𝑋[0,4] b
d
∫a ∫c f(x, y)dydx 4
4
∫ [∫ 2x 2 + 4y dy]dx 0
0
4
∫(2x 2 + 4y) dy = 2yx 2 + 2y 2 │40 = 8(x 2 + 4) 0 4 1 896 ∫ 8(x 2 + 4)dx = 8 ( x 3 + 4x) │40 = 3 3 0
d
b
∫c ∫a f(x, y)dxdy 4
4
∫ [∫ 2x 2 + 4y dx]dy 0
0
4
2 16 ∫ 2x 2 + 4y dx = x 3 + 4xy│40 = (3y + 8) 3 3 0
4
∫ 0
16 16 3 2 896 (3y + 8)dy = ( y + 8𝑦) = 3 3 2 3
z = 2x 2 + 4y
D
R = {(x, y)|a ≤ x ≤ b, 𝜙1 (x) ≤ y ≤ 𝜙2 (x)}
xy
R = {(x, y)|a ≤ x ≤ b, 𝜙1 (x) ≤ y ≤ 𝜙2 (x)}
D
f: D → ℝ D b
f
𝜙2 (x) f(x, y)dxdy 1 (x)
∬D f(x, y)dA = ∫a ∫𝜙
D b
𝜙1 (x) f(x, y)dxdy 2 (x)
∬D f(x, y)dA = ∫a ∫𝜙
𝜙1 (x) ≤ y ≤ 𝜙2 (x)
𝜙2 (x) ≤ y ≤ 𝜙1 (x)
3x 2 + y y=x x=1
y=0
b 𝜙2 (x)
∬ f(x, y)dA = ∫ ∫ [f(x, y)dy]dx D
a 𝜙1 (x)
y xy
𝜙1 𝜙2
𝜙1 (x) = 0 xy
𝜙2 (x) = x y=x x=1
y=
0
x=1 x
x=y
x=0
b
𝜙 (x)
∫a ∫𝜙 2(x) f(x, y)dxdy 1
1
x
1 1 1 1 11 ∫[∫ 3x 2 + y dy]dx = ∫ (3x 2 y + y 2 )|x0 ] dx = ∫ x 2 (6x + 1)dx = 2 2 0 12 0 0
0
D
xy R = {(x, y)|c ≤ y ≤ d, 𝜓1 (y) ≤ x ≤ 𝜓2 (y)} D
f: D → ℝ D
f d 𝜓2 (y)
∬ f(x, y)dA = ∫ ∫ [f(x, y)dx]dy D
c 𝜓1 (y)
D
f(x, y) = 2xy y = −2 y =
2
x = 2 − y ,x = y 𝜓1 , 𝜓2 𝜓1 (y) = y
y = −2 y =
xy 𝜓2 (y) = 2 − y 2
d
𝜓 (y)
∫c ∫𝜓 2(y) [f(x, y)dx]dy 1
1
y
1
∫ [∫ −2
D
2−y2
1
y
[2𝑥𝑦 𝑑𝑥]𝑑𝑦 = ∫ (𝑥 2 𝑦)|2−y2 ] = ∫ − 𝑦(𝑦 4 − 5𝑦 2 + 4) 𝑑𝑦 = − −2
−2
R R = {(x, y)|a ≤ x ≤ b, 𝜙1 (x) ≤ y ≤ 𝜙2 (x)}
9 4
R = {(x, y)|c ≤ y ≤ d, 𝜓1 (y) ≤ x ≤
𝜓2 (y)} D
f: D → ℝ D b 𝜙2 (x)
f
D
d 𝜓2 (y)
∬ f(x, y)dA = ∫ ∫ f(x, y)dxdy = ∫ ∫ [f(x, y)dx]dy D
cosy
2
R
a 𝜙1 (x)
c 𝜓1 (y)
(0,0) (0,1)
f(x, y) = (1,1)
𝜙1 (x) = x
𝜙2 (x) = 1 b 𝜙2 (x) ∫a ∫𝜙 (x) f(x, y)dxdy 1
y x=0
x=1
1
x
1
∫ [∫ cosy 2 dy]dx 0
x
𝜓1 (y) 𝜓2 (y)
x 𝜓1 (y), = 0
𝜓2 (y) = y y y=0
y=1
d 𝜓2 (y) ∫c ∫𝜓 (y) [f(x, y)dx]dy 1 1
y
∫ [∫ cosy 2 dx]dy 0
1
y
1
0
1
y
∫0 [∫0 cosy 2 dx]dy = ∫0 xcosy 2 │0 = ∫0 ycosy 2 𝑑𝑦 = 1 1 1 y 1 ∫ [∫ cosy 2 dy]dx = ∫ [∫ cosy 2 dx]dy = sen(1) 2 0 x 0 0
f
g
∬R[f(x, y) + g(x, y)]dA = ∬R f(x, y)dA + ∬R g(x, y)dA ∬R kf(x, y)dA = k ∬R f(x, y)dA k ∈ ℝ f(x, y) ≥ g(x, y) R = R1 + R 2 m ≤ f(x, y) ≤ M
(x, y) ∈ 𝑅
∬R f(x, y)dA ≥ ∬R g(x, y)dA
∬R f(x, y)dA = ∬R f(x, y)dA + ∬R f(x, y)dA 1
(x, y) ∈ R
2
m ∗ A(R) ≤ ∬R f(x, y)dA ≤ M ∗ A(R)
âˆŹR ecosxseny dA
R = {(x, y)│x 2 + y 2 = 9}
−1 ≤ seny ≤ 1
−1 ≤ cosx ≤ 1
−1 ≤ cosxseny ≤ 1 e−1 ≤ ecosxseny ≤ e1 m = e−1 M = e1
A(R) = π(3)2 = 9π
9Ď€ ≤ âˆŹ ecosxseny ≤ 9Ď€e e đ?‘…