Al-Xoramziy nomidagi Toshkent axborot texnologiyalari universiteti Radio va Mobil aloqa fakulteti 830-20 guruh talabasi Madaminov Jasurbekning Calculas fanidan yozgan

Aylanma jism hajmini hisoblash.

y  f x, x  a, x  b,

y  0

chiziqlar bilan chegaralangan figuraning OX

o’qi atrofida aylanishidan hosil bo’lgan jismning h ajmi

b

b

Vx

  ∫ y ^2 dx   ∫ f ^2  xdx

(7)

a

a

aniq integral bilan hisoblanadi.

x  y,

y  c,

y  d ,

x  0 chiziqlar bilan chegaralangan figuraning

OY o’qi atrofida aylanishidan hosil bo’lgan jismning hajmi

d
d

V y
= 
∫
x 2 dy
=
∫  2 ( y)dy
(8)

c
c

formula bilan hisoblanadi.

8-misol.

y 2

= 2x

parabola,

x  3

to’g’ri chiziq va

OX

o’qi bilan

chegaralangan figuraning OX o’qi atrofida aylanishidan hosil bo’lgan jismning hajmini hisoblang.

Yechish. Masala shartiga ko’ra x o dan 3 gacha o’zgaradi. Demak,

			3		3	3
^Vx	=  ∫ y ^2 dx =  ∫ 2xdx = x ^2								=  (3^2 - 0^2 ) = 9 .
			0		0	0
9-misol.		x 2	+	y 2	= 1 ellipsning OY	o’qi atrofida aylanishidan hosil
		a ^2		b^2