Modul topshirig’i
Nazariy savollar
1.
Darajali qatorlar.
2.
Funksiyalarni Teylor va Makloren qatorlariga yoyish.
3.
Darajali qatorlarning tadbiqlari.
4.
Fur’e qatorlari.
5.
Dekart koordinatalarida ikki o’lchovli integrallarni hisoblash.
6.
Dekart koordinatalarida uch o’lchovli integrallarni hisoblash.
7.
Ikki va uch o’lchovli integrallarda o’zgaruvchilarni almashtirish hamda ularning
tadbiqlari.
8.
Birinchi va ikkinchi tur egrichiziqli integrallar hamda ularning tadbiqlari.
Bajarish uchun vazifalar.
1 – vazifa. Darajaliqatorlar. (1-15). Quyidagi qatorlarni yaqinlashish radiusini toping. (16-
30). Quyidagi qatorlarni yaqinlash ishsohasini toping.
2 – vazifa.
Teylor va Makloren qatorlarigayoying.
3– vazifa. Quyidagi keltirilgan sonlarni berilgan  aniqlikda hisoblang.
4 – vazifa. Funksiyani berilgan oraliqlarda Fur’e qatoriga yoying.
5 – vazifa. Integrallash tartibini o’zgartiring.
6 – vazifa. (1-22) Ikkikarrali integralni hisoblang. (23-30) Berilgan chiziqlar bilan
chegaralangan jismhajmini hisoblang.
7 – vazifa. Uch o’lchovli integrallarni hisoblang.
8 – vazifa. Birinchi tur egri chiziqli integralnihisoblang.
9 - vazifa. Ikkinchi tur egrichiziqli integralni hisoblang.
1.
∑∞
𝑛=1
1 – variant
(𝑥)𝑛
2𝑛
5
2 – variant
3
2.
𝑓(𝑥) = 𝑥 − 4𝑥 + 2𝑥 + 2𝑥 + 1
ko’phadni (𝑥 + 1) ikki hadning darajalari
bo’yicha yoying.
3.
4.
𝑓(𝑥) =
−2𝑥, а𝑔а𝑟 − 𝜋 ≤ 𝑥 < 0 𝑏𝑜’𝑙𝑠а,
Ushbu
{
3𝑥, а𝑔а𝑟 0 ≤ 𝑥 ≤ 𝜋 𝑏𝑜’𝑙𝑠а.
funksiyani Fur’е qаtоrigа yoying.
5.
0
2
 2 y
 dy 
2. 𝑓(𝑥) = 𝑥 − 5𝑥 + 𝑥 2 − 3𝑥 + 4
ko’phadni (𝑥 − 4)ikki hadning darajalari
bo’yicha yoying.
3. cos 36 ,   0, 001
cos18 ,   0, 001
1
𝑛
1. ∑∞
𝑛=1(𝑛 + 1)𝑥
4
3
0
0
1
 y
fdx   dy  fdx.
4. 𝑓(𝑥) = 𝑥 2 funksiyani −𝜋 ≤ 𝑥 ≤ 𝜋
оrаliqdа Fur’е qаtоrigа yoying.
6.
2
2
 15( y  z )dxdydz :
7.
8.
V
 z  x  y , x  y  1;
(V ) : 
 x  0, y  0, z  0.
I   xydl , bu yerda  -uchlari A(-2; 
2), B(6; 1), C(2; 5) nuqtalarda bo‘lgan
uchburchak konturi.
9.
y 2 dx  x 2 dy
I 
, bu yerda  -yarim
x2  y2

0
2
0
 y
1
2 2  sin x
6. 
0
0
 fdx.
 2 y 2
y
 2 dxdy
1
 5 x dxdydz :
2
1 2
1
0 1 ( x  y ) 2 dxdy
1
5.  dy  fdx   dy
V
7.
4 x  3 z  1, y  4;
(V ) : 
 x  0, y  0, z  0.
8. I   x 2 dl , bu yerda  - birinchi

chorakdagi 8 y 2  x 3 egri chiziqning y 2  2 x
parabola bilan ajratilgan qismi.
9. I   (2a  y )dx  (a  y )dy, bu yerda  
sikloidaning birinchi arki:
x  a (t  sin t ), y  a (1  cos t ), 0  t  2
aylana: x  a cos t , y  a sin t , 0  t   .
3 – variant
(𝑥)𝑛
1.
∑∞
𝑛=1
2.
𝑓(𝑥) =
𝑛
1
𝑥+1
funksiyani Makloren
qatoriga yoying.
3.
cos 72 ,   0, 001
4.
𝑓(𝑥) = 𝑥 funksiyani − 𝜋 ≤ 𝑥 ≤
𝜋 оrаliqdа Fur’е qаtоrigа yoying.
4 – variant
𝑛
∑∞
𝑛=1(𝑛𝑥)
2.
𝑓(𝑥) = 𝑙𝑛𝑥 funksiyani 𝑥0 = 1
nuqta atrofida Teylor qatoriga yoying.
1.
3.
sin18 ,   0, 001
5.
1
y
2
2 y 2
 dy  fdx   dy
 fdx.
0
0
1
0
4.
𝑓(𝑥) =
−𝑥, а𝑔а𝑟 − 2 ≤ 𝑥 < 0 𝑏𝑜’𝑙𝑠а,
Ushbu
{
𝑥, а𝑔а𝑟 0 ≤ 𝑥 ≤ 2 𝑏𝑜’𝑙𝑠а.
funksiyani Fur’е qаtоrigа yoying.
4 2
  (3x  2 xy  y )dxdy
2
6.
0 y
y
1
dxdydz
:

x y z
V 1
 
3 4 8
x y z
    1;
(V ) :  3 4 8
 x  0, y  0, z  0.
7.
2 y
1
0
 dy  fdx   dy  fdx.
5.
0
0
1
2 y
0
y
 dx  xdxdy
6.
 (3 x  4 y )dxdydz :
yerda  -
7.
astroida: x 2 / 3  y 2 / 3  a 2 / 3 .
9.
I   y 2 dx  xydy , bu yerda  - A(1;
8.
I   ( x 4 / 3  y 4 / 3 ) dl ,
8.
2
bu
V
 y  x, y  0, x  1;
(V ) : 
2
2
 z  3( x  y ), z  0.
I   ( x  y ) dl , bu yerda  -aylana:


1), B(3; 4) nuqtalarni tutashtiruvchi
to‘g‘ri chiziq kesmasi.

x  y  ax .
2
2
9.
I   yzdx  zxdy  xydz , bu
yerda  -
chizig‘ining
o‘rami:

vint
birinchi
x  R cos t , y  R sin t , z 
5 – variant
1.
∑∞
𝑛=1
(2𝑥−3)𝑛
2𝑛−1
10
5
2.
𝑓(𝑥) = 𝑥 − 3𝑥 + 1 ko’phadni
(𝑥 − 1) ikki hadning darajalari bo’yicha
yoying.
3.
4.
𝑓(𝑥) =
−𝑥, а𝑔а𝑟 − 𝜋 ≤ 𝑥 < 0 𝑏𝑜’𝑙𝑠а,
Ushbu
{ 2
𝑥 , а𝑔а𝑟 0 ≤ 𝑥 ≤ 𝜋 𝑏𝑜’𝑙𝑠а.
funksiyani Fur’е qаtоrigа yoying.
1
5.
 dx
 2
0

 2 x 2
6 – variant
0
∑∞
𝑛=1
2.
𝑓(𝑥) =
1
x
2𝑛
1
𝑥
funksiyani 𝑥0 = 3
nuqta atrofida Teylor qatoriga yoying.
sin 72 ,   0, 001
4.
𝑓(𝑥) = sin𝑥 funksiyani [0, 𝜋] dа
kоsinuslаr bo’yichа Fur’е qаtоrigа
yoying.
1
2
5.
0
fdy   dx  fdy.
(𝑥+1)𝑛
1.
3.
sin 36 ,   0, 001
at
, 0  t  2 .
2
 dy 
0
6.
arcsin y
0
1
fdx   dy
1
2
arccos y
 fdx.
0
 xdxdy D : sikloidaningbirarkasi.
D
x2
2
D y 2 dxdy D : y   x, y  x va y  1
chiziqlarbilanchegaralangansoha.
7.
dxdydz
:
x
y z
V (1 
  )2
16 8 3
x y z
    1;
(V ) : 16 8 3
 x  0, y  0, z  0.
8.
I   ( x 2  z ) dl , bu yerda  chiziq

6.
2
 (27 x  54 y )dxdydz :
V
7.
 y  x, y  0, x  1;
(V ) : 
 z  xy , z  0.

I   ( x  y ) dl , bu yerda  chiziq
8.
x  t, y 

r 2  a 2 cos 2 lemnistikataning o‘ng
I
9.

(1;1;1;)
2
I   ( y  z ) dx  ( z  x ) dy  ( x  y ) dz ,
9.
xdx  ydy  zdz

x 2  y 2  z 2  x  y  2z
buyerda  -aylana:
to‘g‘ri chiziq kesmasi bo‘yicha
hisoblansin.
x 2  y 2  z 2  a 2 , y  xtg , (0    2 ) . Ox
o‘qining musbat qismidan qaraganda
aylana soat millari harakatiga teskari
yo‘nalishda o‘tilsin.
7 – variant
𝑛−1
1. ∑∞
𝑛=1 𝑛! 𝑥
2. 𝑓(𝑥) = √𝑥 3 funksiyani 𝑥0 = 1 nuqta
atrofida Teylor qatoriga yoying.
, z  t 3 egri chiziqning yoyi
(0  t  1) .
yaprog‘i.
( 4; 4; 4 )
3t 2
8 – variant
1. ∑∞
𝑛=1
𝑛!𝑥 𝑛
(𝑛+1)𝑛
𝑥
2. 𝑓(𝑥) = 𝑒 funksiyani Makloren
qatoriga yoying.
1
3. e 2 ,   0, 00001
4. 𝑓(𝑥) = 1 −
𝑥
2
1
3
funksiyani [0, 2] dа
sinuslаr bo’yichа Fur’е qаtоrigа
yoying.
5.
6.
1
2 y
2
0
0
y
1
0
 dy  fdx   dy  fdx.
 y dydx integralni hisoblang bu yerda
3
D
D : y 2  x, y 2  2 x, xy  1va xy  4 chiziqlar
bilan chegaralangan soha.
2
2
 (3x  y )dxdydz :
7.
V
 z  10 y , x  y  1;
(V ) : 
 x  0, y  0, z  0.
3. e ,   0, 00001
4. 𝑓(𝑥) = 𝑥 3 funksiyani [−1; 1] dа
sinuslаr bo’yichа Fur’е qаtоrigа yoying.
1
0
e  ln y
 ln y
0
 y
1
1
5.  dy  fdx   dy  fdx.
6.
 x  y dxdy integralni hisoblang bu
2
2
yerda D : x 2  y 2  x va x 2  y 2  2 x aylanalar
bilan chegaralangan soha.
 (15x  30 z )dxdydz :
7.
V
 z  x 2  3 y 2 , z  0;
(V ) : 
 y  x, y  0, x  1.
8. I   2 y 2  z 2 dl , bu yerda  -aylana:
yerda  -chiziq
I   x 2 yzdl , bu
8.


x2  y2  z 2 , x2  y 2  9
x  y 2  z 2  a2
.

y  x
2
egri
chiziqning
(aylana) yoyi.
9. I   y 2 dx  z 2 dy  x 2 dz , bu yerda
9. I   y 2 dx  x 2 dy , buyerda  : y  x 2
-


chizig‘i:
x  y  z  a , x  y  ax
ning (x,y)
tekislikdan yuqoridagi qismidir.
Vaviana
parabolaningA(0;0), B(2;4) nuqtalari
orasidagi yoyi.
2
2
2
2
9 – variant
2
2
10 – variant
(𝑥)𝑛
(𝑥−1)𝑛
1. ∑∞
𝑛=0 𝑛
1. ∑∞
𝑛=1 2𝑛
2. 𝑓(𝑥) = 2𝑥 funksiyani Makloren
qatoriga yoying.
2 ∙√𝑛+1
1
2. 𝑓(𝑥) =
𝑥
funksiyani 𝑥0 = 2 nuqta
atrofida Teylor qatoriga yoying.
1
3. e 4 ,   0, 00001
3. ln 0, 98,   0, 0001
4. 𝑓(𝑥) = cos 𝑥 funksiyani [0, 𝜋] dа
sinuslаr bo’yichа Fur’е qаtоrigа yoying.
4. 𝑓(𝑥) =
1 + 𝑥, а𝑔а𝑟 − 1 ≤ 𝑥 < 0 𝑏𝑜’𝑙𝑠а,
Ushbu
{
−1, а𝑔а𝑟 0 ≤ 𝑥 ≤ 1 𝑏𝑜’𝑙𝑠а.
funksiyani Fur’е qаtоrigа yoying.
1
2 x 2
0
x2
2
0
1
0
5.  dx  fdy   dx  fdy.
 3
5.  dx
 ( x  3 y  2)ds D : x  y  4
2
2
2
2
2
6. D
0

fdy   dx
 3
 4 x 2
0
 fdy
4 x 2  2
 ( x  xy  2 y )ds D : x  0, y  0 va x  y  1
2
chiziqlar bilan chegaralangan soha.
V
 y  x, y  0, x  1;
7. (V ) : 
 z  xy , z  0.
I   zdl , bu
yerda

x 2  y 2  z 2 , y 2  ax
egri
O(0;0;0)
nuqtasidan
nuqtasigacha bo‘lgan yoyi.
2
6. D
3
 (4  8 z )dxdydz :
8.
0
 (1  2 x )dxdydz :
3
V
7.

-
chiziq
 y  36 x, y  0, x  1;
(V ) : 
 z  xy , z  0.
chiziqning
I   ( 2 x  3 y ) dl , bu yerda  chiziq

A(a;a;a 2 ) 8.
r  a cos 2 lemnistikataning o‘ng yaprog‘i.
I   y 2 dx  x 2 dy , bu
yerda  -chiziqA(- 9. I   ( y 2  z 2 )dx  2 yzdy  x 2 dz , bu yerda  egri


9.
2
3
a;0) nuqtadan B(a;0) gacha bo‘lgan yarim chiziq x  t , y  t , z  t , (0  t  1) .
ellips yoyidir: x  a cos t , y  b sin t .
11 – variant
1.∑∞
𝑛=1
(2𝑥+1)𝑛
12 – variant
5.∑∞
𝑛=1
2𝑛−1
(𝑛+1)5 𝑥 2𝑛
2𝑛+1
3
2. 𝑓(𝑥) = 𝑐𝑜𝑠√𝑥 funksiyani Makloren
qatoriga yoying.
6. 𝑓(𝑥) =
3. 27 ,   0, 001
7. 8 ,   0, 001
4.𝑓(𝑥) = 𝜋 − 𝑥 funksiyani (−𝜋, 𝜋] dа
Fur’е qаtоrigа yoying.
8. 𝑓(𝑥) =

4

sin y
2
cos y
5.  dy  fdx   dy  fdx.
0

0
0
4
funksiyani 𝑥 ning
2−𝑥−𝑥 2
darajalari bo’yicha qatorga yoying.
𝑥2
2
− 1 funksiyani [−3; 3] dа
Fur’е qаtоrigа yoying.
1
0
2
2  x 
9.  dx 
0
0
fdy   dx  fdy
1
3
x
1 x
1 2
  xy ( x  y )dxdy
6. 0 0
10.   xy 2 dxdy
0 x2
10
 (60 y  90 z )dxdydz :
V
 y  4 x, y  0, x  1;
7. (V ) : 
2
2
 z  x  y , z  0.

I 
V
 y  9 x, y  0, x  1;
11. (V ) : 
 z  xy , z  0.
12.
dl
x2  y2
, bu yerda AB-uchlari
AB
8.
A(0;-2), B(4;0) nuqtalardan iborat kesma.
9. I   cos ydx  sin xdy, bu yerda ABAB
5
 ( 3 x  3 )dxdydz :
I   xydl , bu
yerda
x  0, y  0, x  4, y  2 to‘g‘ri
AB
kesma: A(2;2), B(3;4).
13 – variant
(𝑥+1)𝑛
(2𝑛−1)!
-
ushbu
chiziqlardan
tashkil topgan to‘rtburchak konturi.
13. I   ( x 2  y 2 )dx  xydy, bu yerda AB-
kesma: A(2;-2), B(-2;2).
1.∑∞
𝑛=1


14 – variant
𝑛
1. ∑∞
𝑛=1(𝑛 + 1)𝑥
2. 𝑓(𝑥) = 𝑒 −2𝑥 funksiyani Makloren
qatoriga yoying.
2. 𝑓(𝑥) = 𝑥𝑐𝑜𝑠3𝑥funksiyani 𝑥 ning darajalari
bo’yicha qatorga yoying.
3. 24 ,   0, 001
3. sh 0, 3,   0, 0001
4. 𝑓(𝑥) = 1 − 2𝑥 funksiyani [0, 1] dа
kоsinuslаr bo’yichа Fur’е qаtоrigа
yoying.
4.𝑓(𝑥) = 𝑥 + 1 funksiyani (−1; 1] yarim
intеrvаldа Fur’е qаtоrigа yoying.
1
y
e
1
5.  dy  fdx   dy  fdx.
0
0
1
ln y
0
0
 y
1
 2 y
1 3  x
 25 y dxdydz :
2
 (9  18z )dxdydz :
V
 y  4 x, y  0, x  1;
7. (V ) : 
 z  xy , z  0.
dl
8. I   2 2 , bu yerda  - uchlari
x  y 4

O(0;0), A(1;2) nuqtalardan iborat
kesma.
9. I   (4 x  y )dx  ( x  4 y )dy, bu yerda
AB

4
AB yoy y  x egri chiziqning yoyi:
A(1;1) va B(-1;1).
2𝑛
V
 x  15, 3 y  z  1;
7. (V ) : 
 x  0, y  0, z  0.
8. I   ( x  z )dl , bu yerda  egri chiziq

parametrik tenglamalari bilan berilgan:
x  2at 1  t 2 , y  a  ln(1  t 2 ), z  2at 2 ,0  t  1 .
2

2
2
9. I   ydx  xdy , bu yerda AB yoy x  y  1

AB
aylananing A(
√4−𝑥 2
2
;
1
2
) nuqtasidan B (
1
2
;
1
16 – variant
1. ∑∞
𝑛=1
1
1
2
)
nuqtasigacha soat millari harakati yo‘nalishida
o‘tiladi.
15 – variant
(𝑥)𝑛
2. 𝑓(𝑥) =
2
ln y
dxdy
y ( x  3)
2 1
y
1 1 x dxdy
1. ∑∞
𝑛=1
0
6.  
e y
6.
1
5.  dy  fdx   dy  fdx.
funksiyani Makloren
qatoriga yoying.
3. ch 0, 3,   0, 0001
(𝑥−1)𝑛
2𝑛
2. 𝑓(𝑥) = ln(10 + 𝑥) funksiyani 𝑥0 =
1 nuqta atrofida Teylor qatoriga
yoying.
3. sh 0, 5 ,   0, 0001
2
4. 𝑓(𝑥) = cos 𝑥 funksiyani [−𝜋; 𝜋] dа
Fur’е qаtоrigа yoying.
2
4. 𝑓(𝑥) = sin 𝑥 funksiyani [−𝜋; 𝜋]
dа Fur’е qаtоrigа yoying.
1
0
2
0
y
1
0
5.  dy  fdx   dy
 fdx.
y3
2
2 y
0
0
1
0
5.  dy  fdx   dy  fdx.
 2 y 2
6.  ( x 2  y 2 )dxdy, D : x  0, x  1, y  0, y  x 2
6.  ( x  2 y )dxdy, D : y  x 2 , y  5 x  6
D
D
 (8 y  12 z )dxdydz :
dxdydz
:

x y z 4
V (1 
  )
2 4 6
x y z
7.
    1;
(V ) :  2 4 6
 x  0, y  0, z  0.
V
 y  x, y  0, x  1;
(V ) : 
2
2
 z  3 x  2 y , z  0.
8. I   ( x  z ) dl , bu yerda  - uchlari
7.

O(0;0) va A(4;3) nuqtalarni
tutashtiruvchi to‘g‘ri chiziq kesmasi.
9. I   ( x  y ) dx  ( 2 x  y ) dy, bu yerda  -
8. I   ( x  z ) dl , buyerda  :

x  y  z 2  R 2 , y  x aylana yoyining
2
1
2

A(0;0; R), B( R / 2; R / 2; R 2 ) nuqtalar orasidagi
kichik qismi.
9. I   zdx  xdy  ydz , bu yerda  vint
uchlari A(1;1), B(3;3), C(3;-1)
nuqtalarni tutashtiruvchi uchburchak
konturi.

chizig‘ining A(a;0;0) nuqtadan B ( a; 0; 2c )
gacha bo‘lgan bir o‘ramdir:
x  a cos t , y  b sin t , z  ct , 0  t  2 .
1. ∑∞
𝑛=1
(𝑥)𝑛
18 – variant
17 – variant
(𝑥)𝑛
2. 𝑓(𝑥) = arcsin(𝑥)funksiyani 𝑥 ning
darajalari bo’yicha qatorga yoying.
1.∑∞
𝑛=1 𝑛
2. 𝑓(𝑥) = ln(1 + 𝑥) funksiyani 𝑥0 = 1 nuqta
atrofida Teylor qatoriga yoying.
3. sh 0, 37,   0, 0001
3. th 0, 3,   0, 0001
4. 𝑓(𝑥) = 5𝑥 − 4 funksiyani [−𝜋; 𝜋] dа
Fur’е qаtоrigа yoying.
−4, а𝑔а𝑟 − 𝜋 ≤ 𝑥 < 0 𝑏𝑜’𝑙𝑠а,
4.𝑓(𝑥) = { 4, а𝑔а𝑟 0 ≤ 𝑥 ≤ 𝜋 𝑏𝑜’𝑙𝑠а.
Ushbu funksiyani Fur’е qаtоrigа yoying.
3
𝑛
0
2
5.  dx

0
4 x 2 2
6.  e
x  cos y
fdy   dx
3
0
1
 fdy
5.  dy
 4 x 2
dxdy , D : x  0, x   , y  0, y 
D
 ( x  yz )dxdydz :
V
 y  x, y  0, x  1;
7. (V ) : 
2
2
 z  30 x  60 y , z  0.
2

2
0
0
0
 2 y
1
3 y
  fdx   dy  fdx.
6.

 x sin( x  y )dxdy, D : x  0, x   , y  0, y  2
D
8. I  

dl
8 x  y
2
, bu yerda  - uchlari
2
O(0;0) va A(2;2) nuqtalarni tutashtiruvchi
to‘g‘ri chiziq kesmasi.
9. I   ydx  xdy, buyerda  - astroida

x 2 / 3  y 2 / 3  a 2 / 3 ning A(a;0) dan B(0;a)
gacha bo‘lgan yoy.
dxdydz
:
x
y
z 2
V
(1 


)
6
4
16
y
z
x

 1;
7.
 
(V ) :  6
4
16

 x  0, y  0, z  0.

8. I   ydl , bu yerda  : y  2 3 x
2

parabolaning x  2 3 y parabola bilan
kesilgan bo‘lagi.
9. I   xdx  ydy  zdz , bu yerda  : A(a;0;0)
2

dan B ( a; 0; 2b ) gacha bo‘lgan
x  a cos t , y  a sin t , z  bt , 0  t  2 vint
chizig‘ining o‘rami.
19 – variant
𝑛
1.∑∞
𝑛=1(𝑛𝑥)
2. 𝑓(𝑥) =
20 – variant
1.∑∞
𝑛=1
5
6+𝑥−𝑥 2
funksiyani Makloren
qatoriga yoying.
3. th 0, 3,   0, 0001
4. 𝑓(𝑥) = |𝑥| − 2 funksiyani [−𝜋; 𝜋] dа
Fur’е qаtоrigа yoying.
1
x2
2
2 x 2
0
0
1
0
Fur’е qаtоrigа yoying.
x
3
 (5 x  2 z )dxdydz :
V
7. (V ) : 
y  x, y  0, x  1;
 z  x  15 y , z  0.
8. I   x  y dl , bu yerda  : x  y  4 x
aylana yoyi.
x2
4
6.
 z  10(3 x  y ), y  x  1;
(V ) : 
 x  0, y  0, z  0.

2
5.  dx  fdy
D
V
2
1
x
x2
D y 2 dxdy, D : xy  1, y  x, x  2
2
2
 e ydxdy, D : x  0, x  2, y  1, y  e
2
 y dxdydz :
7.
2𝑛−1
1−𝑒 𝑥
2. 𝑓(𝑥) =
funksiyani 𝑥 ning darajalari
𝑥
bo’yicha qatorga yoying.
3. cth 0, 3,   0, 0001
2
4. 𝑓(𝑥) = 𝑥 − 𝑥 funksiyani [−2; 2] dа
5.  dx  fdy   dx  fdy
6.
(2𝑥−3)𝑛
2
2
2
2
8. I   xydl , bu yerda  : 3 x  4 y  12 to‘g‘ri

chiziqning koordinata o‘qlari orasidagi
kesmasi.
I   e y  z dx  e z  x dy  e x  y dz , bu
9. I   dx  dy  dz, buyerda  : A(1;1;1 ) va
9.
B ( 2; 4; 8 ) dan o‘tgan to‘g‘ri chiziq kesmasi.
O (0; 0; 0 )
1
y

1.∑∞
𝑛=1
1
z
1
x
(2𝑥−3)𝑛
2𝑛−1
2. 𝑓(𝑥) =
8+2𝑥−𝑥 2
funksiyani 𝑥0 = 1
𝑛!𝑥 𝑛
(𝑛+1)𝑛
2. 𝑓(𝑥) = ln(1 + 𝑥 − 12𝑥 2 )funksiyani
Makloren qatoriga yoying.
3. arcsin 0, 6 ,   0, 0001
4.𝑓(𝑥) = 2𝑥 funksiyani (0; 1) intеrvаldа
Fur’е qаtоrigа yoying.
2 x
5.  dx  fdy
−2, а𝑔а𝑟 − 𝜋 ≤ 𝑥 < 0 𝑏𝑜’𝑙𝑠а,
4. 𝑓(𝑥) = { 1, а𝑔а𝑟 0 ≤ 𝑥 ≤ 𝜋 𝑏𝑜’𝑙𝑠а.
Ushbu funksiyani Fur’е qаtоrigа yoying.
4
12 x
0
3 x2
5.  dx  fdy
2 x
 ( x  y )dxdy, D : x  0, x  1, y  0, y  x
2
ni birlashtiruvchi
kesma.
1.∑∞
𝑛=1
3. arcsin 0, 4 ,   0, 0001
0
A(1; 3; 5)
22 – variant
nuqta atrofida Teylor qatoriga yoying.
1
bilan
21 – variant
6
yerda  :

2
2
6. D
6.
 xdxdydz , V : x  0, y  0, y  3, z  0, x  z  2
V
dxdydz
:

x y z 2
V (1 
  )
8 3 5
x y z
7.
    1;
(V ) :  8 3 5
 x  0, y  0, z  0.
 10 x dxdydz :
2
V
7.
8. I   xydl , buyerda  : 4 x  3 y  12 to‘g‘ri

chiziqning koordinata o‘qlari orasidagi kesmasi.
9. I   ( y  z ) dx  ( 2  x ) dy  ( x  y ) dz, bu yerda

 -yoy:
x  y  z  25
2
2
2
sferadagi M (3; 4; 0 )
 y  8, 3 x  5 z  1;
(V ) : 
 x  0, y  0, z  0.
8. I   xy ( x  y ) dl , bu yerda  :
x2  y2  R2

aylananingyuqoriyoyi.
y
x

9. I   dx  xdy, bu
yerda  :
y  ln x egri
chiziqning A(0;1) va B (e;1) nuqtalar
orasidagi yoyi.
bilan A(0; 0; 5) ni birlashtiruvchi kata aylananing
eng qisqa yoyi.
23 – variant
24 – variant
(𝑥)𝑛
1.∑∞
𝑛=1 (2𝑛−1)!
1.∑∞
𝑛=1
𝑥
2. 𝑓(𝑥) = 2𝑥 ∙ cos 2 − 𝑥funksiyani 𝑥
2
ning darajalari bo’yicha qatorga yoying.
3. arcsin 0, 9 ,   0, 0001
4. 𝑓(𝑥) = 𝜋 − 2𝑥 , 𝑇 = 2𝜋, [−𝜋; 𝜋].
𝑓(𝑥) funksiyani [0; 𝜋] kеsmаdа juft
dаvоm ettirib Fur’е qаtоrigа yoying.

2
sin x
0
0
6.
 xyzdxdydz , V : x  0, y  0, z  0, x  z  y  1
V
3. arccos 0, 4 ,   0, 0001
𝑥, а𝑔а𝑟 0 ≤ 𝑥 < 1 𝑏𝑜’𝑙𝑠а,
(𝑥)
=
{
𝑇 = 4,
4.
2, а𝑔а𝑟 1 ≤ 𝑥 ≤ 2 𝑏𝑜’𝑙𝑠а.
[0; 4]. 𝑓(𝑥) funksiyani [0; 2] kеsmаdа juft
dаvоm ettirib Fur’е qаtоrigа yoying.
1
2 x2
0
x
6.
 fdy
2
2
 ( x  y )dxdydz , V : z  2, z 
V
x2  y2
2
dxdydz
:
x y z
V 1
 
3 2 8
x y z
    1;
7.
(V ) :  3 2 8
 x  0, y  0, z  0.
2
V
 z  x  3 y , x  2 y  1;
(V ) : 
 x  0, y  0, z  0.
8. I   4 x  y dl , buyerda  :
2
3

 25( y  x)dxdydz :
7.
2𝑛+1
2. 𝑓(𝑥) = √8 − 𝑥 3 funksiyani Makloren qatoriga
yoying.
5.  dx
5.  dx  fdy
(𝑛+1)5 𝑥 2𝑛
2
r  a (1  cos  )

8.
I   y 2 dl ,
bu
:
yerda

kardioidayoyi.
x  a (t  sin t ) , y  a (1  cos t ) , ( 0  t  2 )
2
x
x
y

xe
9. I   ( ye  2 x )dx  e dy, bu yerda  :
sikloidaning bir arkasi.

egri chiziqning
orasidagi yoyi.
va
25 – variant
B (1; e ) nuqtalar
9.
I   y dx  x dy, buyerda  :
2
2

ellipsning A(0; b ) va B (a; 0) nuqtalar
yoyi.
(𝑥)𝑛
1.∑∞
𝑛=0 2𝑛 ∙√𝑛+1
26 – variant
x2 y2

1
a2 b2
orasidagi
1.∑∞
𝑛=1
(𝑥+1)𝑛
2. 𝑓(𝑥) = 𝑥𝑐𝑜𝑠3𝑥 funksiyani 𝑥0 = 3 ning
darajalari bo’yicha qatorga yoying.
(2𝑛−1)!
2. 𝑓(𝑥) = ln(10 + 𝑥) funksiyani 𝑥0 = 0
nuqta atrofida Teylor qatoriga yoying.
3. arc tg 3,   0, 0001
3. arccos 0, 3,   0, 0001
2
4. 𝑓(𝑥) = 𝑥 − 1 funksiyani (−1; 1)
intеrvаldа Fur’е qаtоrigа yoying.
4.𝑓(𝑥) = 3𝑥 − 2 funksiyani (−1; 1)
intеrvаldа Fur’е qаtоrigа yoying.
3
1
8 y 3
0
4 y  4

dxdydz
5.  dy
2x
5.  dx

0
2 x x
fdy
6.
2
x y
2
V
6.
2
2
2
2
V
7.
 4 xydxdydz :
V
 y  6 x, y  0, x  1;
(V ) : 
2
2
 z  3( x  2 y ), z  0.
8. I   ( x 2  y 2  z 2 ) dl , buyerda  :

, V : x 2  y 2  4 y , y  z  4, z  0
2
2
V
2
 (7 x  2 y )dxdydz :
 ( x  y )dxdydz , V : x  y  x, z  0, z  2 x
7.
 fdx
 y  20 x, y  0, x  1;
(V ) : 
 z  3 xy , z  0.
x
dl , buyerda  :
3y  z

I 
8.
t2
t3
x
, y  , z  t chiziqning A(0; 0;0) va
3
2
x  cos t , y  sin t , z  3t , 0  t  2  vint
2 2
B( 2 ;
; 2 ) nuqtalar orasidagi yoyi.
chizig‘ining birinchi o‘rami.
3
2
2
2
9. I   xdy  ydx, bu yerda  : x  y  R 9. I   xdy  ydx, bu yerda  : y  x 2 , x  y 2


aylananing yoyi.
(𝑥−2)𝑛
parabolalar orasidagi egri chiziq yoyi.
27 – variant
1.∑∞
𝑛=0 2𝑛 ∙√𝑛+1
2. 𝑓 (𝑥) = 𝑥𝑠𝑖𝑛𝑥 funksiyani 𝑥0 = 1 ning
darajalari bo’yicha qatorga yoying.
3. 5 1,1,   0,00001
28 – variant
∑∞
𝑛=1
(2𝑥+1)𝑛
2𝑛−1
1.
1
2. 𝑓(𝑥) =
funksiyani 𝑥 ning
√9−𝑥 2
darajalari bo’yicha qatorga yoying.
3. 3 2 ,   0,00001
1, а𝑔а𝑟 − 1 ≤ 𝑥 < 0 𝑏𝑜’𝑙𝑠а,
𝑓(𝑥)
=
{
4.
−𝑥 2 , а𝑔а𝑟 0 ≤ 𝑥 ≤ 1 𝑏𝑜’𝑙𝑠а.
Ushbu funksiyani Fur’е qаtоrigа yoying.
−𝑥 2 , а𝑔а𝑟 − 𝜋 ≤ 𝑥 < 0 𝑏𝑜’𝑙𝑠а,
1 − 𝑥, а𝑔а𝑟 0 ≤ 𝑥 ≤ 𝜋 𝑏𝑜’𝑙𝑠а.
Ushbu funksiyani Fur’е qаtоrigа yoying.
4. 𝑓 (𝑥) = {
3
2y
1
0
0
2 x x

fdy
2
6. x  0, y  0, x  2 y  z  6
 x dxdydz :
2
x  1, y  x, y  2 x, z  x  y , z  x  2 y
2
2
2
2
6.
dxdydz
:
x
y
z
2
V (1 
  )
15 8 4
7.
x y z
    1;
(V ) : 15 8 4
 x  0, y  0, z  0.

1
x2  y2  z2

2 x
5.  dx
5.  dy  fdx
I 
2
dl , bu yerda  : A(1; 2) va
7.
V
 y  3, x  z  1;
(V ) : 
 x  0, y  0, z  0.
8. I   ydl , bu yerda  : y  4 x, x  4 y
2
2

parabolalar orasidagi egri chiziq yoyi.
9. I   xdx  ydy  ( x  y  1)dz, bu yerda  :

A(1;1;1) va B ( 2; 3; 4)
nuqtalarni
tutashtiruvchi to‘g‘ri chiziq.
8.
B (3; 6 ) nuqtalarni tutashtiruvchi to‘g‘ri chiziq
kesmasi.
9.
I   xdy  ydx, buyerda  :

x  4 cos t , y  4 sin t astroidayoyi.
3
3
29 – variant
1.∑∞
𝑛=1
(2𝑥+1)𝑛
1.∑∞
𝑛=1
2𝑛−1
2. 𝑓(𝑥) = 𝑐𝑜𝑠√𝑥 funksiyani Makloren
qatoriga yoying.
3. 27 ,   0,001
4
sin y

2
cos y
5.  dy  fdx   dy  fdx.
0
0

4
(𝑛+1)5 𝑥 2𝑛
2𝑛+1
3
2. 𝑓(𝑥) =
2−𝑥−𝑥 2
funksiyani 𝑥 ning
darajalari bo’yicha qatorga yoying.
3. 8 ,   0,001
4.𝑓(𝑥) = 𝜋 − 𝑥 funksiyani (−𝜋, 𝜋] dа
Fur’е qаtоrigа yoying.

30 – variant
0
4. 𝑓(𝑥) =
𝑥2
2
− 1 funksiyani [−3; 3] dа
Fur’е qаtоrigа yoying.
1
0
2
2  x 
5.  dx 
0
0
fdy   dx  fdy
1
3
x
1 2
  xy ( x  y )dxdy
6. 0 0
1 x
6.   xy 2 dxdy
0 x2
10
 (60 y  90 z )dxdydz :
V
 y  4 x, y  0, x  1;
7. (V ) : 
2
2
 z  x  y , z  0.

I 
dl
x2  y2
V
 y  9 x, y  0, x  1;
7. (V ) : 
 z  xy , z  0.
I   xydl , buyerda  -ushbu
8.
, buyerda AB-uchlari A(0;-
8.
2), B(4;0) nuqtalardan iborat kesma.
9. I   cos ydx  sin xdy, bu yerdaABAB
AB
kesma: A(2;-2), B(-2;2).
5
 ( 3 x  3 )dxdydz :

x  0, y  0, x  4, y  2 to‘g‘ri
chiziqlardan
tashkil topgan to‘rtburchak konturi.
9. I   ( x 2  y 2 )dx  xydy, bu yerda AB-kesma:
AB
A(2;2), B(3;4).