ëÅ‚ öÅ‚
2 3 1 1
xy
íÅ‚
1) x2 + 2xy dyłł dx 2) dx x2y dy
0 0 0 0
ëÅ‚ öÅ‚
0 -4x 3 2x
íÅ‚
1) f(x, y)dyłł dx 2) dx f(x, y)dy
-1 0 2 x
ëÅ‚ öÅ‚
" "
y
1 2x-x2 1
ìÅ‚
3) f(x, y)dy÷Å‚ dx 4) dy f(x, y)dx
íÅ‚ Å‚Å‚
0 0 0 y
x sin(xy)dxdy D = {(x, y) : 1 x 2, 0 y 1}
D
y2
6x2 dxdy D ABC A = (0, 0) B = (0, 4) C = (4, 4)
D
(x + y)dxdy D x = 0 y = 0 x + y - 1 = 0
D
y2 R2 - x2 dxdy D R
D
Å„Å‚
òÅ‚ -1, x < 0
(x2 - y2 + 2) dxdy D = {(x, y) : x2 + y2 4} x = 0, x = 0 ,
ół
1, x > 0
D
|y - x2| dxdy D = {(x, y) : |x| 1, 0 y 2}
D
x2+y2
dxdy D = {(x, y) : x2 + y2 1}
D
ln(x2 + y2)
2
dxdy D = {(x, y) : 1 x2 + y2 }
x2 + y2
D
4 - x2 - y2 dxdy D = {(x, y) : x2 + y2 - 2x 0}
D
(x - 3)2 (y - 5)2
xy dxdy D = (x, y) : + 1
9 25
D
xy3 dxdy D xy = 1 xy = 2 y - x = 0 y - 2x = 0
D
(x + y + 2) dxdy D 1
D
cos(x2 + y2) dxdy D = {(x, y) : x2 + y2 1}
D
x2 + y2 dxdy D (1, 0) 1
D
"
2 4-x2
dx ln 1 + x2 + y2 dy
0 0
y
arctg dxdy D = {(x, y) : 4 x2 + y2 16, -x y x, x 0}
x
D
1 - x2 - y2
dxdy D = {(x, y) : x2 + y2 1, y 0, x 0}
1 + x2 + y2
D
sin x2 + y2 dxdy D = {(x, y) : Ä„2 x2 + y2 4Ä„2}
D
(x + y) dxdy D x2 + y2 = x + y
D
x2 y2 x2 y2
1 - - dxdy D + = 1
a2 b2 a2 b2
D
"
x2 + y2 dxdy D = {(x, y) : 0 x 2, x y 3x}
D
(x + y) dxdy D xy = 1 xy = 2 x - y + 1 = 0 x - y - 1 = 0
D
y = x2 z = x2 + y2 y = 1 z = 0
z = 3 - x2 - y2 x2 + y2 = 3
x2 + y2 + z2 = 1 z = 0
x2 y2 z2
+ + = 1 a, b, c 0
a2 b2 c2
z2 = x2 + y2 x2 + y2 = 2x z = 0
x = 0 y = 0 z" 0 x + y + z - 1 = 0
=
"
y = x y = 2 x z = 0 x + z = 1
x2 + y2 = 1 x2 + z2 = 1
x2 + y2 + z2 = 1 x2 + y2 + 2z = 1
x2 + y2 + z2 = 2az x2 + y2 z2 a > 0
2z = x2 + y2 z = x2 + y2
x + y = a x + y = -a x - y = a x - y = -a a(z - a) = xy z 0 a > 0
z = 1 + x + y z = 0 x = 0 y = 0 x + y - 1 = 0
x + y + z = a z = 0 x = 0 y = 0 x2 + y2 1
z = x + y z = x2 + y2
x2 y2 z2 x2 y2 z2
+ + = 1 + z 0 a, b, c > 0
a2 b2 c2 a2 b2 c2
x2 y2 z2 x2 y2
+ - = -1 + = 1 a, b, c > 0
a2 b2 c2 a2 b2
x2 y2 z x2 y2 x y
+ = + = + z = 0 a, b, c > 0
a2 b2 c a2 b2 a b
x2 + y2 + z2 a2
z2 = x2 + y2
D
y = 0 y = x x2 + y2 - 2x = 0
y = 3 y = x2 - 1
xy = 1 xy = 2 y - x = 0 y - 2x = 0
1
x + y = 1 x + y = 3 y = x y = 2x
2
xy = a2 xy = 2a2 y = x y = 2x x, y > 0
x y x y x y x y
+ = 1 + = 2 = 4 = a, b > 0
a b a b a b a b
x2 y2
+ = 1 a, b 0
a2 b2
3
x2 + y2 2 = 2xy.
r
x2 y2 z2
+ + = 1 a, b, c 0
a2 b2 c2
x2 + y2 + z2 = r2 x2 + y2 = rx r > 0
z = x2 + y2 x2 + y2 = 4
y2 + z2 = x2
x2 + y2 = R2
x2
Á(x, y) = 1 y = y = 2
4
D a Á(x, y) = y
r Á(x, y) = 1
"
-x2
dx
0