"
"
"
 R = {(x, y) : a x b, c y d} P
R1, R2, . . . , Rn
R
"xk, "yk Rk 1 k n

dk = ("xk)2 + ("yk)2 Rk 1 k n
´(P) = max {dk : 1 k n} P
" " " "
¨ = {(x", y1), (x", y2), . . . , (x", yn)} (x", yk) " Rk 1 k n
1 2 n k
P
R P
¨
f P ¨
n

"
f(x", yk)("xk)("yk).
k
k=1
R
f R

n

"
f(x, y)dxdy = lim f(x", yk)("xk)("yk),
k
´(P)0
k=1
R
P R
¨ f R
f R

f(x, y)dÃ.
R
 f g R Ä…, ² " R


Ä…f(x, y) + ²g(x, y) dà = Ä… f(x, y)dà + ² g(x, y)dÃ.
R R R
R R
R1, R2

f(x, y)dà = f(x, y)dà + f(x, y)dÃ.
R R1 R2
f [a, b] × [c, d]
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
b d d b
ðÅ‚ ðÅ‚
f(x, y)dà = f(x, y)dyûÅ‚ dx = f(x, y)dxûÅ‚ dy.
a c c a
[a,b]×[c,d]
îÅ‚ Å‚Å‚ îÅ‚ Å‚Å‚
b d d b
ðÅ‚ ðÅ‚
f(x, y)dyûÅ‚ dx f(x, y)dxûÅ‚ dy,
a c c a
b d d b
dx f(x, y)dy dy f(x, y)dx.
a c c a
1 4 4 1
dx (x2 + y2x)dy, dy (x2 + y2x)dx.
-1 2 2 -1

Ä„ Ä„ Ä„
sin (x + y)dxdy R = - , × 0,
4 4 4
R
xydxdy
, R = [0, 1] × [0, 1]
x2 + y2 + 1
R
f f(x, y) = g(x)h(y) g h
[a, b] [c, d]
ëÅ‚ öÅ‚ ëÅ‚ öÅ‚
b d
íÅ‚ íÅ‚
f(x, y)dxdy = g(x)dxÅ‚Å‚ · h(y)dyÅ‚Å‚ .
a c
[a,b]×[c,d]

xy(x + y)dxdy R = [-1, 1] × [-1, 1]
R

Ä„ Ä„ Ä„
cos(x + y)dxdy R = - , × 0,
4 4 4
R
f
D ‚" R2 R D
f" f R

f(x, y) (x, y) " D,
f"(x, y) :=
0 (x, y) " R\D.
f D

f(x, y)dà := f"(x, y)dÃ,
D R
f D
D Ox
D = {(x, y) : a x b, g(x) y h(x)} ,
 g h [a, b] g(x) < h(x) x " (a, b)
D Oy
D = {(x, y) : p(y) x q(y), c y d} ,
p q [c, d] p(y) < q(y) y " (c, d)
"
y = x2 y = x
y = 0 x = 2 y = x2
Ä„ Ä„
y = |sin x| y = 1 x = - x =
2 2
x2 + y2 = 1
f
D = {(x, y) : a x b, g(x) y h(x)}
Ox
ëÅ‚ öÅ‚
h(x)
b
ìÅ‚
f(x, y)dxdy = f(x, y)dy÷Å‚ dx.
íÅ‚ Å‚Å‚
D a
g(x)
f
D = {(x, y) : p(y) x q(y), c y d}
Oy
ëÅ‚ öÅ‚
q(y)
d
ìÅ‚
f(x, y)dxdy = f(x, y)dx÷Å‚ dy.
íÅ‚ Å‚Å‚
D c
p(y)

(x2 - xy)dxdy D = {(x, y) " R2 : y x, y 3x - x2}
D
(3x - 2y)dxdy D = {(x, y) " R2 : x2 + y2 1}
D

"
xydxdy D = {(x, y) " R2 : y 6 - x, y x, x 0}
D
 D ‚" R2

|D| = dÃ.
D
y = x2 - x y = x
y = ex y = ln x x + y = 1 x = 2
V D ‚" R2
z = f(x, y) z = g(x, y)

|V | = [g(x, y) - f(x, y)] dÃ.
D
x = 0 x = 1 - |y| z = 0 z = 10 - 5x - 2y
z = 5 - 2x - y z = 0 y2 = 3x
z = ey-x x + y = 1 x = 0 y = 0 z = 0