Ilm-fan muammolari yosh tadqiqotchilar talqinida

“Ilm-fan muammolari yosh tadqiqotchilar talqinida” 
mavzusidagi 9-sonli respublika ilmiy konferensiyasi
 
8 
PARAMETRIK KO‘RINISHDA BERILGAN EGRI CHIZIQ UZUNLIGINI 
HISOBLASH 
 
Amirov Javoxir Alisher o‘g‘li 
Guliston davlat universiteti magistranti 
 
Annotatsiya: Ushbu maqolada aniq integral yordamida parametrik ko‘rinishda 
berilgan egri chiziq uzunligini hisoblash keltirilgan. 
Kalit sozlar: parameter, funksiya, Lagranj teoremasi, yoy uzunligi. 
Faraz qilaylik, 
B
A
(
egri chiziq ushbu 
(
)









=
=
t
t
y
t
x
)
(
),
(
tenglamalar sistemasi bilan berilgan bo‘lib, 
1) 
;
]
,
[
)
(
,
]
,
[
)
(






C
t
C
t


2) 
2
1
2
1
,
]
,
[
,
t
t
t
t





uchun 
shartlarning bajarilishi bilan birga 
)
(
),
(
t
t


funksiyalari 
]
,
[


da uzluksiz 
)
(t


hamda 
)
(t


hosilalarga ega bo‘lsin. 
]
,
[


segmentning ixtiyoriy 


)
...
(
,...,
,
1
0
1
0


=



=
=
n
n
t
t
t
t
t
t
P
bo‘laklashini 
olib, 
ularga 
mos 
B
A
(
yoyiniig 
)
,
(
k
k
k
k
y
x
A
A
=
))
(
,
)
(
(
k
k
k
k
t
y
t
x


=
=
nuqtalarini bir-biri bilan to‘g‘ri chiziq kesmasi yordamida 
birlashtirishdan hosil bo‘lgan l  siniq chiziq perimetri 

−
=
+
+
−
+
−
=
1
0
2
1
2
1
))]
(
)
(
[
)]
(
)
(
[
)
(
n
k
k
k
k
k
t
t
t
t
l





ni qaraymiz. 
Lagranj teoremasidan foydalanib topamiz: 
(
)
k
k
k
k
n
k
k
k
n
k
k
k
k
k
k
k
t
t
t
t
t
t
t
t
l
−
=




+

=
=
−


+
−


=
+
−
=
−
=
+
+


1
1
0
2
2
1
0
2
1
2
2
1
2
)
(
)
(
)
(
)
(
)
(
)
(
)
(










---

“Ilm-fan muammolari yosh tadqiqotchilar talqinida” 
mavzusidagi 9-sonli respublika ilmiy konferensiyasi
 
9 
bunda 
.
]
,
[
.
]
,
[
1
1
+
+


k
k
k
k
k
k
t
t
t
t


Keyingi tenglikni quyidagicha yozib olamiz: 
+



+

=

−
=
k
n
k
k
k
t
l
1
0
2
2
)
(
)
(
)
(





k
k
k
n
k
k
k
t



+

−

+

+

−
=
]
)
(
)
(
)
(
)
(
[
2
2
1
0
2
2








(2) 
bunda, 
].
,
[
1
+

k
k
k
t
t

Madomiki, 
]
,
[
)
(
)
(
2
2




C
t
t


+

ekan unda 
]
,
[
)
(
)
(
2
2




R
t
t


+

bo‘lib, 


−
=
→

+

=



+

1
0
2
2
2
2
0
)
(
)
(
)
(
)
(
lim
n
k
k
k
k
dt
t
t
t
p









(3) 
bo‘ladi. 
Ixtiyoriy 
d
c
b
a
,
,
,
haqiqiy sonlar uchun ushbu 
d
b
c
a
d
c
b
a
−
+
−

+
−
+
2
2
2
2
tengsizlik o‘rinli bo‘ladi [1]. 
Haqiqatan ham, 
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
(
)
(
)
.
a
c
a
c
b
d
a
b
c
d
a
c
a
b
c
d
a
b
c
d
b
d
b
d
a
c
b
d
a
b
c
d
+
−
+
−
+
−
+
=
 − 
+
+
+
+
+
+
+
+
− 
 − + −
+
+
+
Bu tengsizlikdan foydalanib topamiz: 



+

−

+


−
=
k
k
k
n
k
k
k
t
]
)
(
)
(
)
(
)
(
[
2
2
1
0
2
2









---

“Ilm-fan muammolari yosh tadqiqotchilar talqinida” 
mavzusidagi 9-sonli respublika ilmiy konferensiyasi
 
10 


−
=
−
=



−

+


−


1
0
1
0
)
(
)
(
)
(
)
(
n
k
n
k
k
k
k
k
k
k
t
t










−
=
−
=



+




1
0
1
0
.
)
(
)
(
n
k
n
k
k
k
t
t




]
,
[
)
(
,
]
,
[
)
(






R
t
R
t




bo‘lganligi sababli 
0
]
)
(
)
(
)
(
)
(
[
lim
2
2
1
0
2
2
0
=


+

−
−

+


−
=
→
k
k
k
n
k
k
k
t
p









(4) 
bo‘ladi [2]. 
(3) va (4) munosabatlarni e’tiborga olib, 
0
→
p

da (*) tenglikda limitga o‘tsak, 
u holda 
B
A
(
yoyining uzunligi uchun 
dt
t
t
B
A


+

=





)
(
)
(
)
(
2
2
(
(5) 
bo‘lishi kelib chiqadi. Bu formula yordamida yoy uzunligi hisoblanadi. 
Misol. Ushbu 
)
0
(
)
cos
1
(
)
sin
(



−
=
−
=
t
t
a
y
t
t
a
x
tenglamalar sistemasi bilan berilgan 
B
A
(
egri chiziqning (tsikloidaning) uzunligi 
topilsin. 
Yechish: Ravshanki, 
2
2
2
2
2
2
2
2
2
( )
(1 cos ),
( )
sin ,
( )
( )
(1 cos )
sin
2(1 cos ),
( )
( )
2(1 cos )
x t
a
t
y t
a
t
x
t
y
t
a
t
a
t
a
t
x
t
y
t
a
t


=
−
=


+
=
−
+
=
−


+
=
−
bo‘ladi. (5) formulaga ko‘ra izlanayotgan egri chiziqning uzunligi 
a
t
a
dt
t
a
dt
t
a
B
A
8
)
2
(cos
4
2
sin
2
)
cos
1
(
2
)
(
2
0
2
0
2
0
=

−
=
=
−
=






(
bo‘ladi. 
 
 

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“Ilm-fan muammolari yosh tadqiqotchilar talqinida” 
mavzusidagi 9-sonli respublika ilmiy konferensiyasi
 
11 
Foydalanilgan adabiyotlar: 
1. 
Sh.A.Alimov “Matematik analiz”, Тоshkent., 2011. 
2. 
A.Gʻoziyev, I.Isroilov, M.Yaxshiboyev “Matematik analizdan misol va 
masalalar”, Тоshkent., “Yangi asr avlodi”– 2006. 

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“Ilm-fan muammolari yosh tadqiqotchilar talqinida” 
mavzusidagi 9-sonli respublika ilmiy konferensiyasi
 
12 
THE FUNCTION OF THE RAILS. THE PROCEDURE FOR RE-USE OF 
RAILS USED FOR A LONG YEAR 
Shermatov Diyorbek Akmaljon ugli 
Tashkent State transport University 
Uralov Akmal Shakar ugli 
Tashkent State transport University 
Annotation: This article lists the task to be put on the rail and the procedure for 
reusing previously used rails. 
Keywords: Rail functions, grouping according to the years of use of the rails. 
The purpose of rails is to create surfaces with the lowest resistance for rolling 
wheels of rolling stock, directly perceive and elastically transfer the impact of wheels 
on supports (sleepers, bars) and guide the wheels of rolling stock. In addition, at 
sections with autoblocking (automatic system of train traffic control, in which the run 
between stations is divided into one or more block sections, usually from 1 to 3 km 
long. At the beginning of each block section there is an automatically operating 
through traffic light), railroad strands serve as conductors of signal current, and on 
electrified sections - reverse traction current. [1]. 
Rails must be sufficiently strong, durable, reliable in operation, hard and 
simultaneously ductile (non-fragile), as they take shock-dynamic loads from the 
wheels of rolling stock. Weight of rails, cross-section profiles, chemical composition 
of rail steel and manufacturing technology are interrelated and together determine the 
operational qualities of the rail as an element of the track superstructure. [1]. 
The possibility of reuse of old-age rails removed from the track is determined 
before their removal from the track. The suitability groups of old-age rails to 
determine the possibility and scope of reuse in the track of old-age rails and rails 
strands removed from the track in all types of repairs, continuous or single 
replacement and ongoing maintenance. Rails suitability group is established at the 

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