РГР [вариант 10]

		2				2		r4				2		4 cosϕ				2		(80 cos				2					4		2	ϕ) dϕ
=		∫			5r		−		sin				ϕ	|		dϕ =		∫							ϕ −64 cos					ϕ sin
								4
	−π													0			−π
	2																2
π											π																π
2									1		2										1				sin 2ϕ					π
													(1 +cos 2ϕ) dϕ =														2
∫				2	ϕ dϕ =						∫												ϕ +
	cos																										|	=
									2												2				2					2
−π											−π																−π
2											2																2

ππ

2			4	ϕ sin	2		2		1 +cos 2ϕ 2				1 −cos 2ϕ				dϕ =
∫ cos						ϕ dϕ = ∫				2				2
−π							−	π
2								2
		π
=	1	∫2		(1 + 2 cos 2ϕ +cos2 2ϕ) (1 −cos 2ϕ) dϕ =
	8
		−	π
			2
		π
	1	2		(1 +cos 2ϕ −cos								2ϕ) dϕ =		1			sin 2ϕ
=		∫					2	2ϕ −cos			3					ϕ +			−
	8													8			2
		−	π
			2

ππ

−		1			∫2			(1 +cos 4ϕ) dϕ −												1	∫2 (1 −sin2 2ϕ) d (sin 2ϕ)=
	16																			16
					−		π														−	π
							2															2
			1							sin 2ϕ						1						sin 4ϕ					1			sin	3 2ϕ		π
																																	2
=					ϕ +										−					ϕ +						−			sin 2ϕ −				|	=
			8								2					16						4					16				3
																																	−π
																																	2
			ϕ + sin							3	2ϕ	− sin 4ϕ								π			π
=																				2	=
																				|
		16							48						64				−π			16
																				2
V = 80								π	−64				π		= 36π
								2				16

6z = x				2	+ y		2							2			2						6z = x2 + y2										x2	+ y2 =12
													= z		+ y
												6z
		2π			2 3				16−r2			2π			2 3									r	2
V = ∫dϕ						∫ r dr ∫						dz = ∫dϕ			∫	r			16 −r2				−			dr =
		0				0				r2		0			0								6
									6
	2π					12											12	r	3				2π					1		3				r	4		12
= ∫dϕ						∫		16 −r2 d (16 −r2 )− ∫												dr		=	∫dϕ				−		(16	−r2 )		−					|	=
																															2
																		6										3						24
	0					0											0						o													0
=	2∫π 38		dϕ =					76	π = 79,59
	0	3						3

(x −1)2 + y2 = −2x + 2

2π	1	47−44r cos ϕ	2π	1	2π	1
V = ∫ϕ∫r dr		∫
			dz = ∫dϕ∫r (22 −22r2 )dr = ∫dϕ (11r2 −5,5r4 )|				=
0	0	22(r2 −2r cosϕ+1)+3	0	0	0	0

=2∫π 5,5π dϕ =11π

0

												2	4−r											2		r
M = ∫ dϕ ∫r dr ∫ 6z dz = ∫								dϕ ∫r dr 3z					|		= ∫		dϕ ∫3r					4 −r			−		dr =
																										4
	0	0		r / 2			0	0					r / 2			0			0
π / 2	2		4r	−	5r3		π / 2		2r	2	−		5r4		2	=	π / 2	33		dϕ =			33π
= ∫	dϕ ∫	3			4	dr =	∫	dϕ 3					16		|		∫	4					8
0	0						0							0			0