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“C”guruh
Integrallarni hisoblang.
9.69.
dx
x
x
5
4
3
16
9.80.
5
1
3
1
x
xdx
9.81.
2
1
1
2
2
4
dx
x
x
9.82.
9
4
1
x
dx
x
9.83.
2
1
ln
2
e
e
dx
x
x
9.84.
4
0
1
2
x
dx
9.85.
dx
x
x
1
2
2
2
1
9.86.
8
ln
3
ln
1
x
x
e
dx
e
9.88.
а
)
2
0
cos
xdx
x
в
)
dx
xe
x
2
ln
0
9.88.
dx
x
x
e
1
ln
9.89.
dx
x
e
1
2
ln
9.80.
dx
arctgx
3
0
9.81.
dx
e
x
x
1
1
2
9.82.
dx
x
e
x
0
sin
9.83.
dx
x
x
2
/
0
sin
3
9.84.
dx
x
x
x
cos
sin
9.85.
dx
x
x
2
/
0
3
sin
cos
9.86.
7
7
6
4
2
sin
dx
x
x
x
9.88.
dx
x
2
/
0
3
sin
9.88.
dx
e
x
2
ln
0
1
9.89.
1
0
2
5
4
x
x
dx
9.90.
0
cos
2
3
x
dx
9.91.
3
1
2
1
5
x
x
x
dx
9.92 .
2
/
1
0
1
1
dx
x
x
9.93.
2
1
2
2
x
x
dx
9.94.
0
1
3
1
1
x
dx
9.95.
4
/
0
3
xdx
tg
9.96.
dx
x
x
3
3
2
2
9
9.98
2
9
4
1
t
x
x
dx
9.98
8
3
1
x
xdx
9.99
2
1
2
1
x
x
dx
9.100
3
1
2
2
1
dx
x
x
9.101
1
0
x
x
x
e
e
dx
e
9.102
29
3
3
2
3
2
2
3
2
x
dx
x
9.103
6
3
4
2
9
dx
x
x
9.104
1
2
/
2
2
2
4
dx
x
x
9.105
a
x
a
x
dx
0
2
2
9.106
2
/
0
cos
2
x
dx
§ 9.2. Yassi figuralar yuzlarini hisoblash.
1) Uzluksiz
0
x
f
x
f
y
egri chiziq
a
x
va
b
x
to’g’ri chiziqlar hamda Ox o’qining [a,b] kesmasi
bilan chegaralangan egri chiziqli trapetsiyaning yuzi
b
a
dx
x
f
S
(9.13)
formula bilan hisoblanadi (9.1-chizma).
9.1- chizma.
Agar [a,b] kesmada
0
x
f
va uzluksiz bo’lsa, u holda aABb yuzasi
b
a
dx
x
f
S
(9.14)
formula bilan topiladi.
2) Uzluksiz
0
y
y
x
egri chiziq,
1
c
y
va
d
y
to’g’ri chiziqlar hamda Oy o’qining [c,d] kesmasi
bilan chegaralangan egri chiziqli trapetsiyaning yuzi
d
c
dy
y
S
(9.15)
formula bilan hisoblanadi .
3) Uzluksiz
x
f
y
1
va
x
f
y
2
egri chiziqlar hamda
b
a
b
x
a
x
,
to’g’ri chiziqlar bilan
chegaralangan sohaning yuzi
b
a
dx
x
f
x
f
S
1
2
, (9.16)
formula bilan hisoblanadi (9.2 - chizma).
9.2 – chizma
.
4) Agar [d,e] kesmada
x
f
y
funksiya uzluksiz va chekli sonda o’z ishorasini almashtirsin (9.3 - chizma). Masalan,
[a,b], [b;d] musbat va [d,e] kesmada manfiy qiymatlarni qabul qilsin.
9.3 – chizma
.
S
A
B
a
b
x
y
0
x
f
y
S
a
b x
y
0
x
f
y
1
x
f
y
2
+S
1
-
S
2
S
3
y
x
a
b
d
e
U holda
e
d
d
b
b
a
dx
x
f
dx
x
f
dx
x
f
S
S
S
S
3
2
1
.
5) Yuqori chegarasi parametrik ko’rinishda berilgan egri chiziq
t
t
y
t
x
,
va yon tomonlari
x
a
va
x
b
chiziqlar bilan chegaralangan egri chiziqli trapetsiya yuzasini hisoblash uchun (9.13)
formuladan foydalanamiz.
dt
t
t
t
b
x
t
a
x
dt
t
dx
t
x
t
y
ydx
dx
x
f
S
b
a
b
a
6) Qutb koordinatalar sistemasida berilgan egri chiziqli OAB sektorning yuzini (9.4-chizma) quyidagi formula
yordamida topamiz:
2
1
2
2
0
2
1
2
1
lim
d
r
r
S
(9.18)
9.4 – chizma
.
“A” guruh
Quyidagi chiziqlar bilan chegaralangan figuralarning yuzini toping
.
9.107.
0
,
4
2
y
x
y
9.108
1
2
2
2
2
b
y
a
x
9.109
h
x
px
y
,
2
2
9.110
0
,
2
3
2
y
x
x
y
9.111
0
,
4
,
1
,
4
y
x
x
xy
9.112
0
,
,
ln
y
e
x
x
y
9.113
0
,
4
2
2
x
x
y
9.114
0
,
8
,
3
2
x
y
x
y
9.115
0
,
4
3
2
x
x
y
9.116
0
4
3
2
2
x
x
y
egri chiziq ilmog’i
9.117
2
2
2
,
x
y
x
y
9.118
4
,
4
2
x
y
x
x
y
9.119
x
a
x
y
a
2
3
2
2
9.120
1
,
3
2
x
x
x
y
9.121
2
2
2
a
x
x
a
y
atrofidagi ilmog’i
9.122
a
x
a
x
e
e
a
y
2
zanjir chiziq
a
x
va
0
y
9.123
t
a
y
t
a
x
cos
1
,
sin
sikloidaning bir davri (arkasi) va OX o’qi
“B”guruh
y
x
0
B
A
r
2
1
r
r
9.124
t
a
y
t
a
x
3
3
sin
,
cos
astroida
9.125
2
cos
2
2
a
r
limneskata 9.126
cos
1
a
r
kardioida
9.128
2
4
x
x
y
parabola va O
x
o’qi bilan chegaralangan.
9.128.
2
2
/
;
;
e
y
e
y
e
y
x
x
9.129.
0
;
2
2
4
y
x
x
y
9.130.
1
;
2
3
2
x
y
x
x
y
9.131.
0
;
2
;
4
;
3
2
x
y
xy
x
y
9.132.
x
x
y
x
y
3
2
;
2
3
9.133 .
0
;
1
;
1
y
x
y
x
y
9.134
.
4
/
;
0
;
0
;
2
cos
x
x
y
x
y
9.135
.
2
4
;
2
x
y
x
y
9.136
.
0
;
3
;
;
1
2
y
x
x
y
xy
9.138.
2
2
;
2
;
2
;
0
x
x
y
y
x
x
x
9.138.
2
/
;
0
;
2
arcsin
y
x
x
y
9.139.
0
;
0
16
2
3
;
;
1
2
2
x
y
x
y
x
x
y
9.140.
1
;
1
2
2
x
y
x
y
9.141.
0
;
2
;
4
2
2
y
x
x
y
x
y
(2-chorak)
9.142.
0
;
0
4
2
;
2
2
/
1
2
y
y
x
x
y
(1-chorak)
§ 9.3. Yoy uzunligini hisoblash.
1) To’g’ri burchakli Dekart koordinatalari sistemasida berilgan egri chiziq yoyining uzunligi [a;b] kesmada
x
f
y
tenglama bilan berilgan. Egri chiziq uchun
x
f
uzluksiz bo’lsa, u holda uning yoyi uzunligi (9.5 - chizma).
b
a
dx
x
f
l
2
1
(9.18)
formula bilan hisoblanadi.
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