Å„Å‚ Å„Å‚
òÅ‚x = r cos Õ òÅ‚r > 0
r
óły = r sin Õ ółÕ " (0, 2Ä„)
Å„Å‚ Å„Å‚
ôÅ‚x = r cos ¸ cos Õ ôÅ‚r > 0
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
òÅ‚ òÅ‚
r2 cos ¸
ôÅ‚y = r cos ¸ sin Õ ôÅ‚Õ " (0, 2Ä„)
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
ółz = r sin ¸ ół¸ " (-Ä„ , )
Ä„
2 2
Å„Å‚ Å„Å‚
ôÅ‚x = r cos Õ ôÅ‚r > 0
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
òÅ‚ òÅ‚
r
ôÅ‚y = r sin Õ ôÅ‚Õ " (0, 2Ä„)
ôÅ‚ ôÅ‚
ôÅ‚ ôÅ‚
ółz = z ółz " R
Ä„/2
2 2x y
dy
dx; cos(x + y)dx dy
(1+x+y)2
0 0 0
x/2
1 1 e2 ln y
f(x, y)dy dx; f(x, y)dx dy
0 x 1 0
f(x, y) dxdy
D
1
f(x, y) = D = [1, 2] × [3, 4]
(y+x)2
f(x, y) = yex D A = (1, 1), B = (2, 2), C = (3, 2)
f(x, y) = sin x cos y D A = (a, 0), B = (0, a), C = (0, 0)
a > 0
f(x, y) = 2y + x D A = (0, 0), B = (2, 2), C = (1, 1)
1
f(x, y) = 2xy2 D y = x, y = , y = 2
x
1
f(x, y) = D y = x, y = x2
1+x
4
f(x, y) = y + x D y = x, y = 4x y =
x
f(x, y) = e-2x D 2 - x2 d" y d" x2 - 3x
1 1
f(x, y) = xy2 D y = , y = -x, y " [1, 2]
x
f(x, y) = 2xy D y2 + x2 d" 16, x d" -2
"
1 1
f(x, y) = D |y| = , x = e, x = e
ln2 x x
 f(x, y) = cos(xy) D 2y |x| = Ä„, y = 1, y = 2
f(x, y) = xy D xy = 1, xy = 2, y = x2, y = 3x2
" "
" "
f(x, y) = x + y x = 0, y = 0, x + y = 2.
ey
f(x, y) = D y = ln |x| , y = 0, y = 1
(4+x)2
|x|
4
f(x, y) = x + 2y D y = |x| , y = , y =
4 x
1
"
f(x, y) = D 1 d" y2 + x2 d" 4
x2+y2
f(x, y) = x2 + y2 D x2 + y2 - 2x d" 0
f(x, y) = x2 + y2 D x2 + y2 d" 4, y e" |x|
f(x, y) = x D x2 + (y - 1)2 = 1, y = x, (x e" y)
1
"
f(x, y) = D (x2 + y2)2 - 4x(x2 + y2) = 4y2,
x2+y2
x2 + y2 = 4, (x e" 0)
f(x, y) = (x2 + y2)3 D 2y d" x2 + y2 d" 6y
f(x, y) = sin x2 + y2 D 4Ä„2 d" x2 + y2 - 2x d"
8Ä„2, y e" 0
f(x, y) = x + y D 0 d" x d" y, x2 + y2 d" 18
Å»
f(x, y) = x2 + y2 D K((1, 0), 1)
Å» Å» Å»
K((-1, 0), 1) K((0, 1), 1) K((0, -1), 1)
f(x, y) = x x2 + y2 D (x2 + y2)2 = x2 - y2, x = 0,
(x e" 0)
D ‚" R2
" (x, y) " D (x, -y) " D.
f : D R
" (x, y) " D f(x, y) = -f(x, -y).
f(x, y) dxdy = 0
D
f : [a, b]×[c, d] R f(x, y) dxdy
[a,b]×[c,d]
f(3x, 4y) dxdy = 2
a b c d
[ , ]×[ , ]
3 3 4 4
f(x, y, z) dxdydy
D
"
f(x, y, z) = y(cos x + z) D y = x y = 0
Ä„
z = 0 x + z =
2
f(x, y, z) = z x2 + y2 D z = -2 z = 3
x2 + y2 = 1 x2 + y2 = 16
f(x, y, z) = xy D z = 0 x = 0 y = 0
2x + y + z = 6
f(x, y, z) = z2 D x2 + y2 = z2 x2 + y2 = -2x
f(x, y, z) = (x + y)2 D z = -2 z = 7 - x2 - y2
"
f(x, y, z) = z D y2 d" x d" 6 - y 0 d" z d" x
1
"
f(x, y, z) = D z = -1 z = 1
x2+y2+(z-2)2
x2 + y2 = 1
f(x, y, z) = x2 + y2 D z = 0 x2 + y2 + z2 = 1
x2 + y2 + z2 = 16
f(x, y, z) = z D 2x + y = 2 x = 0 y = 0
z = 0 x2 + y2 + z2 = 9
f(x, y, z) = y D z = y, z = 0, y = 1 - x2
f(x, y, z) = x2yz D x = 2, y = -x, y = x2,
z = 0, z = x + y
f(x, y, z) = x2 + y2 D z = x2 + y2, z = 1,
z = 4;
1
"
f(x, y, z) = D x2 + y2 + z2 = 4,
x2+y2+z2
x2 + y2 + z2 = 16;
f(x, y, z) = x2+y2 D x2+y2 = 4, z = x2 + y2,
z = 0;
f(x, y, z) = x2 + y2 + z2 D x2+y2+z2-z = 0
z = x2 + y2 y = 0 z = 0 x = 0 x + y = 4
x2 + y2 = 1 y2 + z2 = 1
 x2 + y2 = 2y x2 + y2 + z2 = 4
z = x2 + y2 y2 + x2 = z2
x2 + y2 + z2 = 8 y2 + x2 = 2z
z = x2 + y2 2y2 + x2 = 1 y = x y = 2x
z = 4 + x2 + y2 z = -2 y2 + x2 = 4
z = 4 + x2 + y2 z = -2 y2 + x2 + 4y = 0
y2 = z z = -1 y = 0 y = 3 - x y = 3 + x
z = 1 + 4y - x2 - y2 z = 1
"
4
z = x2 + y2 z = 1 z = 2
x2 + y2 + z2 = 9, z = x2 + y2;
z = 4x2 + y2, z = 4 - 3y2;
y2 + z2 = 1, y = x, x = 0.
z2 = 4x y2 = 4x x = 1
x2 + y2 + z2 = 4 3z = y2 + x2
z = 2x + 2y - 13 y2 + x2 = 2x
"
z = 2xy y = 1 y = x x = 4
z = 9 - x2 - y2 x2 + y2 = 1 x2 + y2 = 4
x2 + z2 = y2 x2 + y2 = 4
V = [0, a] × [0, b] × [0, c] a, b, c > 0 Á(x, y, z) = x + y + z
Å»
V = K((0, 0, 0), 3) Á(x, y, z) = x2 + y2 + z2