97
Fаn–tехnikа tаrаqqiyоti jаmiyаtning mоddiy vа mа’nаviy еhtiyоjlаrini hаr tоmоnlаmа
qоndirish uchun kеng imkоniyаtlаr оchib bеrаyоtgаn bо’lsа, ikkinchi tоmоndаn tаbiiy
rеsurslаrdаn tоbоrа intеnsiv fоydаlаnishni tаqоzо qilmоqdа. Bu jаrаyоn tеvаrаk–аtrоfdаgi
muhitgа, еkоlоgik muvоzаnаtgа sаlbiy tа’sir kо’rsаtmоqdа. Shuning uchun hаm rеspublikаmiz
tаrаqqiyоtining hоzirgi bоsqichidа tаbiiy rеsurslаrdаn оqilоnа fоydаlаnish, о’rmоn, tоzа hаvо,
hаyvоnоt dunyоsi, yеr–suv, еkоlоgik tоzа mаhsulоtlаr ishlаb chiqаrish tо’g’risidа g’аmхо’rlik
qilish hаr birimizning muqаddаs burchi bо’lmоg’i kеrаk.
Fоydаlаnilgаn аdаbiyоtlаr ro’yhati:
1. Абдулкосимов А.А., Боймирзаев К.М. Проблемы оптимизации экологического
состояния антропогенных ландшафтов Средней Азии.// Еducаtiоn аnd Sciеncе fоr
Sustаinаblе Dеvеlоpmеnt. Intеrnаtiоnаl Thеоrеticаl аnd Prаctiаl Cоnfеrеncе. 6-8 аpril, Tаshkеnt,
Uzbеkistаn.-2016, 42-43 с.
2. Ходжиматов А.Н. Воҳа ландшафтлари: мазмуни ва агрогеосистема тушунчалари. –
Тошкент, 1993, Китоб-1. – 64 б.
3. Аkаbоyеv I.Z., Qаrоrоv Z.R. Iqlim о’zgаrishi nаtijаsidа yuzаgа kеlgаn аyrim gеоеkоlоgik
muаmmоlаr. “Янги Ўзбекистонда география фани ва таълимидаги муаммолар”
мавзусидаги республика илмий – амалий конференцияси материаллари, Жиззах – 2022,
431-433 bеtlаr.
AYLANISH SIRTINING YUZINI HISOBLASH USULLARI
Xoljigitov Dilmurod Xolmurod o‘g‘li
Xolmanova Klara Yangiboy qizi
O‘zbekiston Milliy universiteti Jizzax filiali “Amaliy matematika” kafedrasi
o‘qituvchilari
Annotatsiya.
Ushbu
maqolada aylanish sirtlarining yuzini hisoblash usullari keltirilgan
va unga doir misollar ham berilgan.
Kalit so‘zlar.
Aylanish sirtlari, differensiallanuvchi funksiya, O
x
o‘qi atrofida
aylantirish natijasida hosil qilingan sirt, Gorizontal aylanma sirt yuzasini hisoblash, Vertikal
aylanma sirt yuzasini hisoblash.
Aytaylik, [
a;b
] kesmada manfiy bo‘lmagan uzluksiz differensiallanuvchi
y=f(x)
funksiya
grafigini O
x
o‘qi atrofida aylantirish natijasida sirt hosil qilingan bo‘lsin (1-rasm). Shu aylanish
sirtining yuzini hisoblashda quyidagi formuladan foydalanamiz:
dx
x
f
x
f
S
b
a
2
)]
(
'
[
1
)
(
2
(1)
Bu
aylanish sirtining yuzini topish formulasidir.
1-rasm
Gorizontal aylanma sirt yuzasini hisoblash
:
> SurfaceOfRevolution(f(x), x=a..b, output=integral);
a
1
i
x
i
x
i
x
i
s
b
x
98
Vertikal aylanma sirt yuzasini hisoblash
:
> SurfaceOfRevolution(f(x), x=a..b, output=integral, axis=vertical);
1 misol.
x
y
funksiya grafigining x
[1;2] kesmaga mos qismini Ox o‘qi atrofida
aylantirishdan hosil bo‘lgan sirtning yuzini hisoblang.
Yechish.
x
x
f
x
x
f
2
1
)
(
'
,
)
(
. Bularni (1) ga qo‘yamiz.
2
1
2
1
2
4
1
2
2
1
1
2
dx
x
dx
x
x
S
3
3
2
1
3
4
1
1
4
1
2
3
4
4
1
3
2
2
x
.
283
,
8
6
5
5
27
8
5
5
8
27
3
4
>restart; with(plots): with(Student[Calculus1]):
>f:=x->sqrt(x):
>plot(f(x),x=1..2,color=red,style=line, thickness=2, title=`YOY`);
> SurfaceOfRevolution(f(x), x=1..2, output=integral);
> SurfaceOfRevolution(f(x), x=1..2);
> evalf(%);
283154665
> SurfaceOfRevolution(f(x), x=1..2, output=plot);
K
5
6
5
p C
9
2
p
a
b
2
p
|
f
(
x
)
|
0
d
d
x
f
(
x
)
1
2
C
1
d
x
∫
a
b
2
p
|
x
|
0
d
d
x
f
(
x
)
1
2
C
1
d
x
∫
∫
1
2
p
x
1
C
4
x
x
d
x
99
2-misol.
2
x
y
funksiya grafigining x
[1;2] kesmaga mos qismini:
O
y
o‘qi atrofida aylantirishdan hosil bo‘lgan sirtning yuzini hisoblang.
>restart; with(plots): with(Student[Calculus1]):
>f:=x->x^2:
>plot(f(x),x=1..2,color=red,style=line, thickness=2, title=`YOY`);
> SurfaceOfRevolution(f(x), x=1..2, output=integral, axis=vertical);
> SurfaceOfRevolution(f(x),x=1..2,axis=vertical);
> evalf(%);
30.84648972
> SurfaceOfRevolution(f(x), x=1..2, output=plot,axis=vertical);
3-misol.
2
3
2
,
2
x
y
x
y
chiziqlar bilan chegaralangan yassi figyrani O
y va Ox
o’qiga
atrofida aylanishsan hosil bo’lgan sirtning yuzasini hisoblang.
1)
Berilgan
chiziqlar bilan chegaralangan yassi figyra:
> restart; with(plots):
with(Student[Calculus1]):
> f2:=x->2-x^2: f1:=x->(x^2)^(1/3):
> plot({f2(x),f1(x)},x=-2..2, y=0..2,color=[red,blue],
style=line,thickness=2,title=`YUZA`);
2)
figyrani Oy o’qiga atrofida aylanishsan hosil bo’lgan sirtni qurish:
K
5
6
5
p C
17
6
17
p
∫
1
2
2
p
x
4
x
2
C
1
d
x
100
> S1:=SurfaceOfRevolution(f2(x), x=0..1, output=plot, axis=vertical):
S2:=SurfaceOfRevolution(f1(x), x=0..1, output=plot, axis=vertical):
display({S2,S1}, axes=boxed, title=`SIRT`);
3)
figyrani Oy o’qiga atrofida aylanishsan hosil bo’lgan sirtning yuzasini hisoblash
> SurfaceOfRevolution(f2(x), x=0..1, output=integral,
axis=vertical)+SurfaceOfRevolution(f1(x), x=0..1, output=
integral,
axis=vertical);
> SurfaceOfRevolution(f2(x), x=0..1,axis=vertical)+
SurfaceOfRevolution(f1(x), x=0..1,axis=vertical);
> evalf(%);
369550289
4)
figurani Ox o’qiga atrofida aylanishsan hosil bo’lgan sirtni qurish
.
> S1:=SurfaceOfRevolution(f2(x), x=0..1, output=plot):
S2:=SurfaceOfRevolution(f1(x), x=0..1, output=plot):
display({S2,S1}, axes=boxed, title=`SIRT`);
5)
figurani Ox o’qiga atrofida aylanishsan hosil bo’lgan sirtning yuzasini hisoblash
.
> SurfaceOfRevolution(f2(x), x=0..1, output=integral)+
SurfaceOfRevolution(f1(x), x=0..1, output=integral);
1
6
p
( 5 5
K
1 )
C
28
81
p
13
K
8
243
p
ln( 2 )
C
8
243
p
ln( 3
C
13 )
∫
0
1
2
p
x
4
x
2
C
1
d
x
C
ó
õ
0
1
2
3
p
x
4
C
9
x
(
2
/
3
)
x
(
2
/
3
)
d
x
101
> SurfaceOfRevolution(f2(x), x=0..1)+
SurfaceOfRevolution(f1(x), x=0..1);
> evalf(%);
171163206
3-misol.
3
/
2
3
/
2
3
/
2
a
y
x
astroida-chizig’i bilan chegaralangan egri chiziqli yassi yuzani O
x
o’qi atrofida aylanishdan hosil bo’lgan sirt yuzasini hisoblash.
Yechish:
O
x
o’qi atrofida aylanishdan hosil bo’lgan sirt yuzasini hisoblash:
dt
y
x
y
S
x
2
2
)
(
)
(
2
formulasiga asosida, astroidaning 101arametric tenglamasi
x
=
a
cos
3
(t), y=
a
sin
3
(t) (
t
)ga
asosan
dt
y
x
y
S
x
2
2
)
(
)
(
2
2
=
2
/
0
2
4
2
2
4
2
3
2
2
cos
sin
9
sin
cos
9
sin
4
)
(
)
(
2
2
dt
t
t
a
t
t
a
t
a
dt
y
x
y
S
x
=
5
12
0
2
/
sin
5
12
)
(sin
sin
12
cos
sin
3
sin
4
2
5
2
2
/
0
4
2
2
/
0
2
a
t
a
t
d
t
a
dt
t
t
a
t
a
1)
astroida grafigini quramiz
:
> with(plots):
> plot([1*cos(t)^3,1*sin(t)^3, t=0..2*Pi],thickness=2);
2)
astroidani
[
0,π/2
]
kesmasiga mos qimining Ox o’qiatrofida aylanishdan hosil bo’lgan sirt
grafigi
.
> restart; with(plots):
with(Student[Calculus1]):
> VolumeOfRevolution(sqrt(1^(2/3)-x^(2/3))^3, x=0..1, output=plot);
33
16
p
ln( 2 )
C
23
16
p
5
C
33
32
p
ln
0
1
2
C
1
4
5
1
C
128
1215
p
C
494
1215
13
p
∫
0
1
K
2
p
(
K
2
C
x
2
)
4
x
2
C
1
d
x
C
∫
0
1
2
3
p
x
(
2
/
3
)
4
C
9
x
(
2
/
3
)
x
(
2
/
3
)
d
x
102
3)
astroida-chizig’I bilan chegaralangan figurani
[
0,π/2
]
kesmasiga mos qismini Ox
o’qiatrofida aylanishdan hosil bo’lgan sirt yuzasini hisoblab, uni ikkiga ko’paytirib, to’la sirtni
topamiz.
> with(Student:-Calculus1):
> x:=t->a*cos(t)^3; y:=t->a*sin(t)^3
> SS1:=(2*Int(2*Pi*y(t)*sqrt(diff(x(t),t)^2+diff(y(t),t)^2), t=0..Pi/2));
> value(%);
> evalf(%);
Foydalanilgan adabiyotlar
1.
Xoljigitov, Dilmurod. "GEOMETRİYANING ALGEBRAIK TENGLAMALARNI
YECHISHGA BAZI TATBIQLARI."
Matematika va kompyuter fanlari jurnali
1.3 (2021).
2.
Dilmurod X., Jo'raboyevich RN AXBOROT TEXNOLOGIYALARINING MULTIMEDIA
VOSITALARIDAN
MATEMATIKA
FANINI
O'QITISH
JARAYONIDA
FOYDALANISHNING AHAMIYATI //International Journal of Contemporary Scientific and
Technical Research. – 2022. – B. 708-711.
3.
Klara
X.
et
al.
MURAKKAB
TUZILISHDAGI
ARALASH
MAKSIMUMLI
DIFFERENSIAL TENGLAMALAR SISTEMALARI UCHUN CHEGARAVIY SHART
//Лучшие интеллектуальные исследования. – 2023. – Т. 10. – №. 5. – С. 34-40.
4.
Алимов С., Холманова К. Matematika fanini o ‘qitish jarayonida zamonaviy axborot
texnologiyalaridan foydalanish //Информатика и инженерные технологии. – 2023. – Т. 1. –
№. 2. – С. 565-569.
x
:=
t
/
a
cos(
t
)
3
y
:=
t
/
a
sin(
t
)
3
12
5
p
a
2
csgn(
a
)
7.539822370
a
2
csgn(
a
)
SS1
:=
2
∫
0
1
2
p
6
p
a
sin
(
t
)
3
(
a
2
cos
(
t
)
4
sin
(
t
)
2
C
a
2
sin
(
t
)
4
cos
(
t
)
2
)
1
2
d
t
