79
.
cos
sin
.
20
.
cos
.
19
.
sin
.
18
.
cos
.
17
.
sin
.
16
.
cos
1
sin
.
15
.
.
14
.
.
13
3
2
5
5
4
4
2
3
3
3
xdx
x
xdx
xdx
xdx
xdx
dx
x
x
xdx
ctg
xdx
tg
.
dx
x
x
3
3
2
sin
cos
.
21
.
.
5
cos
3
sin
.
24
.
2
sin
4
sin
.
23
.
3
cos
7
cos
.
22
dx
x
x
xdx
x
xdx
x
26-mashg’ulot. Aniq integral va uni hisoblash
Mustaqil bajarish uchun topshiriqlar
Ushbu integrallarni hisoblang.
.
cos
.
10
.
4
.
9
.
3
.
8
.
4
5
.
7
.
4
2
.
6
.
21
4
.
5
.
1
.
4
.
cos
.
3
.
1
.
2
.
)
(
.
1
2
0
0
2
2
2
5
2
0
1
2
6
1
2
2
2
4
2
1
4
0
9
4
2
3
xdx
x
dx
x
x
x
x
x
x
dx
x
x
dx
x
x
dx
dx
x
x
a
dt
dx
x
dx
x
x
e
27-mashg’ulot. Aniq integralning tatbiqlari
Mustaqil bajarish uchun topshiriqlar
1. Qo’yidagi chiziqlar bilan chegaralangan figuralarning yuzlarini hisoblang.
.
0
,
,
ln
)
3
;
0
,
4
)
2
;
0
,
8
6
)
1
2
2
y
e
x
x
y
x
y
x
y
x
x
y
4)
2
2
x
y
parabola,
3
,
1
x
x
to’g’ri chiziqlar va
OX
o’qi bilan chegaralangan;
;
,
3
)
8
;
4
,
4
)
7
;
,
2
)
6
;
0
,
2
)
5
3
3
2
2
2
2
2
2
t
t
y
t
x
x
y
x
x
y
x
y
x
y
x
y
y
x
2.
0
,
4
,
1
,
4
y
x
x
xy
chiziqlar bilan chegaralangan figuraning
OX
o’qi
atrofida aylanishdan hosil bo’lgan jism hajmini hisoblang.
3. 1)
0
)
4
(
3
2
x
va
x
y
chiziqlar bilan chegaralangan figuraning
OY
o’qi
atrofida aylanishidan hosil bo’lgan jism hajmini hisoblang.
2)
8
,
0
,
3
y
x
x
y
chiziqlar bilan chegaralangan figuraning
OY
o’qi atrofida aylanishidan
hosil bo’lgan jism hajmini hisoblang.
4. O’zgaruvchan kuchning bajargan ishi aniq integral yordamida qanday hisoblanadi?
5. Mehnat unumdorligi funksiyasi nima?
6. Ishlab chiqarish mehnat unumdorligini aniq integral yordamida hisoblash mumkinmi va
qanday?
7. Omborga keltirilgan mahsulotlar miqdorini aniq integral yordamida qanday hisoblanadi?
80
8. Mahsulot ishlab chiqarish arifmetik progressiya bo’yicha o’suvchi bo’lsa,
uning hajmi aniq
integral yordamida qanday hisoblanadi?
9. Yillik daromad funksiyasi nima?
10. Diskontli daromad nima va u aniq integral yordamida qanday hisoblanadi.
28-mashg’ulot. Aniq integralni taqribiy hisoblash. Xosmas integrallar
Mustaqil bajarish uchun topshiriqlar
1.
1
0
2
dx
x
integralni
5
n
bo’lakka bo’lib, trapesiyalar
formulasi bilan taqribiy hisoblang. Uning aniq qiymati va taqribiy qiymati farqini baholang.
2.
2
1
1
x
dx
integralni
10
n
teng bo’laklarga bo’lib,
trapesiyalar va Simpson
formulalari yordamida taqribiy hisoblang ikkala holda ham xatolarni baholang.
3.Quyidagi integrallarning yaqinlashuvchiligini tekshiring.
.
)
6
;
)
5
;
)
4
;
)
3
;
)
2
;
)
1
0
2
2
1
2
0
0
0
0
3
2
dx
e
x
x
x
dx
xe
dx
e
x
dx
x
dx
x
x
x
4. Quyidagi integrallarning yaqinlashuvchiligini tekshiring.
1
0
1
0
1
6
2
2
0
2
0
3
2
2
2
.
ln
)
6
;
1
)
5
;
1
)
4
;
)
1
(
)
3
;
)
1
(
)
2
;
)
4
(
)
1
e
x
x
dx
dx
x
x
dx
x
dx
x
dx
x
dx
29-mashg’ulot. Ko’p o’zgaruvchili funksiyalar
Mustaqil ish uchun topshiriqlar
1. Quyidagi funksiyalarning aniqlanish sohasini aniqlang va uning qandayligini izohlang:
2
2
9
1
)
1
y
x
u
;
2
2
3
2
1
)
2
y
x
u
;
;
)
ln(
)
3
y
x
u
2
2
2
4
)
4
z
y
x
u
;
xy
z
)
5
;
)
6
x
y
xy
z
2. Quyidagi limitlarni hisoblang.
xy
xy
y
x
.
4
2
lim
)
1
0
0
;
;
)
sin(
lim
)
2
2
0
x
xy
y
x
3.
Quyidagi funksiyalarning istalgan nuqtada uzluksizligini ko’rsating.
2
2
2
2
2
2
3
2
)
4
;
2
)
3
;
3
)
2
;
)
1
z
y
x
z
y
x
u
y
x
z
y
x
z
4 Quyidagi funksiyalarning uzilish nuqtalarini toping.
.
)
3
;
)
2
;
2
6
)
1
2
2
2
2
2
2
2
2
y
x
y
x
z
y
x
xy
z
y
x
z
30-mashg’ulot. Ikki o’zgaruvchili funksiyaning xususiy hosilasi va to’la differensiali
Mustaqil bajarish uchun topshiriqlar
1. Quyidagi funksiyalarning xususiy hosilalarini toping:
81
3
2
3
3
)
1
y
y
x
x
z
z
x
y
z
x
y
u
y
x
xy
z
)
3
;
)
2
;
.
arcsin
1
)
4
2
y
y
x
xy
y
x
z
2. Quyidagi funksiyalarning to’la differensiallarini toping:
;
ln
)
1
2
2
y
x
x
z
z
y
x
u
2
)
2
;
2
2
)
3
y
x
z
;
;
3
2
)
4
2
2
2
z
y
x
u
.
ln
)
5
t
x
s
3.
xy
z
funksiya uchun
)
4
;
5
(
0
P
nuqtada
2
,
0
,
1
,
0
y
x
bo’lganda
z
va
dz
larni hisoblang.
4.
0
0
02
,
2
59
cos
32
sin
)
2
;
)
04
,
1
(
)
1
taqribiy hisoblang.
5.
3
3
y
y
x
z
funksiyaning ikkinchi tartibli xususiy hosilalarini toping.
6.
s
x
t
ln
1
1
funksiya uchun
2
2
2
2
1
s
x t
s
x
x
ekanligini tekshiring.
7.
)
2
(
t
x
arctg
u
bo’lsa,
2
2
2
2
0
U
x
U
x t
bajarilishini tekshiring.
8.
2
2
2
)
(
y
x
z
ikkinchi tartibli xususiy hosilalarni toping.
9.
u
x
y
z
1
2
2
2
funksiya
2
2
2
2
2
2
0
u
x
u
y
u
z
tenglamani qanoatlantirishini isbotlang.
10.
2
4
y
x
u
ikkinchi tartibli to’la differensialini toping.
11.
z
x
y
sin cos
ikkinchi tartibli to’la differensialini toping.
12
;
)
1
2
2
x
y
u
x
y
x
u
ln
)
2
ikkinchi tartibli to’la differensiallarini toping.
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