f(x, y) = x3 + xy2 + 6xy + 1.
z = 6 - x2 - y2, z = x2 + y2.
-xy2dy + x2ydx,
K
K x2 + y2 = 1
y
y + = x.
x
x
y + 5y + 6y = .
f(x) = x [-Ä„, Ä„]
"
f(x, y) = y x - y2 - x + 6y.
z = 6 - x2 - y2, z = x2 + y2.
(2x3 - 11y)dx + (4x + sin y)dy,
K
K x2 + y2 = 16
"
xy - 4y = x2 y
y(1) = 1
2x
y + 2y + 2y = + 1.
f(x) = |x| [-Ä„, Ä„]
f(x, y) = x3 + xy2 + 6xy + 1.
x2 + y2 + z2 9, x2 + y2 4, z 0.
-xy2dy + x2ydx,
K
K x2 + y2 = 1
2y ln x
y + = y2 , y(1) = 4.
x x
2x
y - 4y + 4y = x .
f(x) = x2 [-Ä„, Ä„]
"
y = arctgx, x " [0, 1]
Ox
z = x2 + y2 + 4, z = 6 - x2 + y2.
-xydx + (y2 + 1)dy,
C
C
D = (x, y) " R2 : 0 y x2, 0 x 1 .
2yy - y2 - x = 0.
y + 4y = 1 + x.
0 - Ä„ < x < 0
f(x) = .
1 0 x < Ä„
1
y =
x2 + 4x + 5
x " [0, 1]
z = 3 - x2 - y2, z = 2 x2 + y2.
x2
y + y =
y2
y(0) = 1
y + 4y = 2 cos 3x.
ex(1 - cos y)dx - ex(1 - sin y)dy,
C
C y = sin x y = 0 x " [0, Ä„]
Å„Å‚
ôÅ‚
òÅ‚0 x " (-Ä„, 0)
Ä„
f(x) = 2 x " 0,
ôÅ‚
ół0 x " Ą , 2
0
2
"
f(x, y) = y x - y2 - x + 6y.
x2 + y2
z = 4 - x2 - y2, z = " .
3
xy - y = x2e-2x
y(1) = 1
y + 9y = x2 + 3.
(4,2)
2xydx + x2dy,
(0,0)
y = x2
(1 - x2)dx + x(1 + y2)dy
L
L
"
y = x + 4, y = 0, x = 0.
f(x, y) = x3 + 3xy2 + 12xy.
V = {(x, y, z) " R3 : x2 + y2 + z2 4, x2 + y2 3z}.
y + y = xy-6
y(0) = 2
y + 3y = e-3x.
x(y + z)dl,
L
Å„Å‚
ôÅ‚
ôÅ‚x = cos t
òÅ‚
y = sin t , t " [0, Ä„]
L : .
ôÅ‚
ôÅ‚z = 3t
ół
4
xydx - xdy
L
L
"
y = x, y = 0, x " [1, 2].
1
y = , y = 0, x " [-2, 1].
x2 + 4x + 13
y
arctg dxdy,
x
D
D = {(x, y) : 4 x2 + y2 16, -x y, y x}
x
y + y =
y
y(0) = 1
y - 6y + 10y = cos 2x.
xdx - x2dy,
K
K x2 + y2 = 8x
Ä„
1 x " 0,
4
f(x) =
Ä„
2 x " , Ä„
4
"
y = xe-2x
Ox 0 x 1
ln(x2 + y2) dxdy,
D
D = {(x, y) : r2 x2 + y2 R2, r, R > 0}
x
y + xy =
y3
y(0) = 2
y + 4y + 5y = 2 sin 2x.
y(x - y)dx + xdy,
K
K y2 = 4x A(1, -2) B(0, 0)
Ä„
0 x " 0,
2
f(x) =
Ä„
2 x " , Ä„
2
1
y = , y = 0, x " [0, 1].
x2 + 4x + 8
xy dxdy,
D
D = {(x, y) : x2 + y2 2x}
(x + y2)dx + 2xydy = 0
y(1) = 2
y + 3y = 3-3x.
y(x - y)dx + xdy,
K
K (0, 0)
(2, 0) (1, 1)
Ä„
0 x " 0,
2
f(x) =
Ä„
2 x " , Ä„
2
x = a(cos t + t sin t) Ä„
K : t " 0, .
2
y = a(sin t - t cos t)
z = x2 + y2, z = 6 - x2 - y2.
x2
y + y = ,
y
y(0) = 1
y + 5y = 2 cos 3x.
ex(1 - cos y)dx - ex(1 - sin y)dy,
C
C y = sin x y = 0 x " [0, Ä„]
Å„Å‚
ôÅ‚
òÅ‚0 x " (-Ä„, 0)
Ä„
f(x) = 2 x " 0,
ôÅ‚
ół0 x " Ą , 2
Ä„
2
"
x2+y2
xy · e
,
x2 + y2
D
D = {(x, y) : x2 + y2 4, y |x|}
z = 8 - x2 - y2 + z2, z = 2 + x2 + y2, x2 + y2 = 1, x = 0, y = 0
x2 + y2 1 x 0 y 0
x2 + y2dl,
L
Å„Å‚
ôÅ‚
òÅ‚x = cos t + t sin t
L : y = sin t - t cos t, t " [0, Ä„] .
ôÅ‚
ółz = ln 5
arctgy y arctgx x
+ dx + + + 2y dy,
1 + x2 x 1 + y2 y
K
K y = x A(1, 1) B(2, 2)
(x2 + y2)dxdz,
S+
1
S+ y = a2 - x2 - z2 y = a2
2
"
3n
3n + 1 1
(-1)narctg · · xn.
3n e
n=1
 Ox
1
y = "
x2 - 1
2 x y
z2 = x2 + y2, x2 + y2 = 4, z 0.
1 1
y + y = y2.
x x
y - 5y - 7y = sin x.
xy
ln(1 + y)dx - dy,
1 + y
C
C A(0, 0) B(2, 0) C(0, 4)
1 x " [-Ä„, 0)
f(x) = .
x x " [0, Ä„)
x2 y3
f(x, y) = + - 2y2 - xy + 6y.
2 3
x = e2t cos t ln 2
L : t " 0, .
2
y = e2t sin t,
B = (x, y, z) " R3 : y 0, x2 + y2 1, 0 z arctg x2 + y2 .
y x
y - = .
x x2 + 1
y - 4y + 3y = x2 + 1.
x2ydy - x2dx,
C
C A(1, 1) B(1, -1)
C(3, -1) D(3, 1)
-x
y = x x 0
Ox
z 9 - x2 - y2 + z2, (x - 1)2 + y2 1, z = 0.
yzdx + zxdy + xydz,
L
at
L x = r cos t y = r sin t z =
2Ä„
z = 0 z = a a, r > 0
yxy-1dx + xy ln xdy = 0.
-3x
y + 6y + 9y = .
"
n(x - 1)n
.
n2 + 3n
n=1
1
y = , y = 0.
x2 + 2x + 10
z 9 - x2 - y2, 4x2 + y2 1, z -1.
y · y + y2 = ex.
y + 4y + 4y = e-2x.
(y2 + 2y)dx - (ey + 2x)dy,
C
C (x - 1)2 + y2 = 1
2 x " [-Ä„, 0)
f(x) = .
-x x " [0, Ä„)
"
-x
y = x - 1 ·
Ox
f(x, y) = 2x3 + xy2 + 5x2 + y2.
z = 1 + x2 + y2, z = 9 - x2 - y2.
C : x2 + y2 = 4x,
yxy-1dx + xy ln xdy = 0.
y + 4y + 5y = cos 2x.