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Высшая математика

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Кузнецов_математика

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∫∫∫y2ch(xy) dx dy dz;

	V
4.21.	x	= 0, y	= −1, y = x,
	x	= 0, y	= −1, y = x,
	V	= 0, z	= 2.
	z	= 0, z	= 2.
		π
	∫∫∫x2cos		xy dx dy dz;
4.23.	V	2
4.23.	V	= 2, y	= x, y = 0,
	x	= 2, y	= x, y = 0,
	V	= 0, z	= π.
	z	= 0, z	= π.
	∫∫∫x2cos(π xy) dx dy dz;
	V
4.25.	x	=1, y	= 2x, y = 0,
	x	=1, y	= 2x, y = 0,
	V	= 0, z	= 4π.
	z	= 0, z	= 4π.
	∫∫∫y2ch(3xy) dx dy dz;
	V
4.27.	x	= 0, y	= 2, y = 6x,
	x	= 0, y	= 2, y = 6x,
	V	= 0, z	= −3.
	z	= 0, z	= −3.
	∫∫∫x2sin(4π xy) dx dy dz;
	V
4.29.	x	=1, y	= x 2, y = 0,
	x	=1, y	= x 2, y = 0,
	V	= 0, z	= 8π.
	z	= 0, z	= 8π.
	∫∫∫x2sh(xy) dx dy dz;
	V
4.31.	x	= 2, y	= x 2, y = 0,
	x	= 2, y	= x 2, y = 0,
	V	= 0, z	=1.
	z	= 0, z	=1.

Задача 5. Вычислить.

∫∫∫x2 z ch(xyz) dx dy dz;

	V
4.22.	x	=1, y =1, z =1,
	x	=1, y =1, z =1,
	V	= 0, y = 0, z = 0.
	x	= 0, y = 0, z = 0.
	∫∫∫y2 zcos				xyz		dx dy dz;
	∫∫∫y2 zcos						dx dy dz;
	V		9
4.24.	V
	x = 9, y =1, z = 2π ,
	V	= 0, y = 0, z = 0.
	x	= 0, y = 0, z = 0.
				xyz
	∫∫∫y2 z ch							dx dy dz;
	∫∫∫y2 z ch							dx dy dz;
4.26.	V					2
4.26.	V
	x = 2, y = −1, z = 2,
	V	= 0, y = 0, z = 0.
	x	= 0, y = 0, z = 0.
	∫∫∫2y2 z ch(2xyz) dx dy dz;
	V
4.28.		=	1			= 2, z = −1,
	x			, y
	x			, y
	V		2
		= 0, y = 0, z = 0.
	x	= 0, y = 0, z = 0.
	∫∫∫8y2 z e− xyz					dx dy dz;
	V
4.30.	x	= 2, y = −1, z = 2,
	x	= 2, y = −1, z = 2,
	V	= 0, y = 0, z = 0.
	x	= 0, y = 0, z = 0.

∫∫∫x dx dy dz;

5.1. V : y =10x, y = 0, x =1, z = xy, z = 0.

∫∫∫15(y2 + z2 ) dx dy dz;

5.3.V : z = x + y, x + y =1, x = 0, y = 0, z = 0.

∫∫∫(1+ 2x3 ) dx dy dz;

5.5. V : y = 9x, y = 0, x =1,

z = xy, z = 0.

∫∫∫y dx dy dz;

5.7. V : y =15x, y = 0, x =1, z = xy, z = 0.

∫∫∫(3x2 + y2 ) dx dy dz;

5.9.V : z =10y, x + y =1, x = 0, y = 0, z = 0.

∫∫∫(4 + 8z3 ) dx dy dz;

5.11. V : y = x, y = 0, x =1,

z = xy, z = 0.

∫∫∫V

dx dy dz

;

z 4

5.2. V :

=1,

x = 0, y = 0, z = 0.

∫∫∫(3x + 4y) dx dy dz;

5.4. V : y = x, y = 0, x =1,

z = 5(x2 + y2 ), z = 0.

∫∫∫(27 + 54y3 ) dx dy dz;

5.6. V : y = x, y = 0,

x =1,

z =

xy, z = 0.

∫∫∫V

dx dy dz

;

5.8. V :

=1,

x = 0, y = 0, z = 0.

∫∫∫(15x + 30z) dx dy dz;

5.10. V : z = x2 + 3y2 , z = 0, y = x, y = 0, z = 0.

∫∫∫(1+ 2x3 ) dx dy dz;

5.12. V : y = 36x, y = 0, x =1, z = xy, z = 0.

∫∫∫21xz dx dy dz;

5.13. V : y = x, y = 0, x = 2, z = xy, z = 0.

∫∫∫(x2 + 3y2 ) dx dy dz;

5.15.V : z =10x, x + y =1, x = 0, y = 0, z = 0.

∫∫∫	10	x +	5	dx dy dz;
	3		3

5.17. V : y = 9x, y = 0, x =1, z = xy, z = 0.

∫∫∫3y2 dx dy dz;

5.19. V : y = 2x, y = 0, x = 2, z = xy, z = 0.

∫∫∫x2 dx dy dz;

5.21.V : z =10(x + 3y), x + y =1, x = 0, y = 0, z = 0.

∫∫∫63(1+ 2y ) dx dy dz;

5.23. V : y = x, y = 0, x =1, z = xy, z = 0.

∫∫∫

5.14. V :

		dx dy dz						;

	+		x	+	y	+	z 6
1
		10			8
							3

x + y + z =1, 10 8 3

x = 0, y = 0, z = 0.

∫∫∫(60y + 90z) dx dy dz;

5.16. V : y = x, y = 0, x =1,

z = x2 + y2 , z = 0.

∫∫∫(9 +18z) dx dy dz;

5.18. V :

∫∫∫

5.20. V :

y = 4x, y = 0, x =1, z = xy, z = 0.

dx dy dz

;

=1,

x = 0, y = 0, z = 0.

∫∫∫(8y +12z) dx dy dz;

5.22. V : y = x, y = 0, x =1,

z = 3x2 + 2y2 , z = 0.

∫∫∫(x + y) dx dy dz;

5.24. V : y = x, y = 0, x =1,

z = 30x2 + 60y2 , z = 0.

∫∫∫V

dx dy dz

;

5.25. V :

=1,

x = 0, y = 0, z = 0.

∫∫∫y2 dx dy dz;

5.27.V : z =10(3x + y), x + y =1, x = 0, y = 0, z = 0.

∫∫∫(x2 + 4y2 ) dx dy dz;

5.29.V : z = 20(2x + y), x + y =1, x = 0, y = 0, z = 0.

∫∫∫x2 z dx dy dz;

5.31. V : y = 3x, y = 0, x = 2, z = xy, z = 0.

∫∫∫xyz dx dy dz;

5.26. V : y = x, y = 0, x = 2, z = xy, z = 0.

∫∫∫	5x +	3z	dx dy dz;

		2
V

5.28. V : y = x,

y = 0, x = 2,

z = x2 +15y2 , z = 0.

∫∫∫V

dx dy dz

;

5.30. V :

=1,

x = 0, y = 0, z = 0.

Задача 6. Найти площадь фигуры, ограниченной данными линиями.

6.1.y = 3x, y = 4ex , y = 3, y = 4.

6.2.x = 36 − y2 , x = 6 − 36 − y2 .

6.3.x2 + y2 = 72, 6y = −x2 (y ≤ 0).

6.4.x = 8 − y2 , x = −2y.

6.5. y = 3 , y = 8ex , y = 3, y = 8. x

6.6. y = x , y = 1 , x =16.

22x

6.7.x = 5 − y2 , x = −4y.

6.8. x2 + y2 =12, -6y = x2 ( y ≤ 0).

6.9.y = 12 − x2 , y = 23 − 12 − x2 , x = 0 (x ≥ 0).

6.10.y = 3 x, y = 3 , x = 9.

22x

6.11.y = 24 − x2 , 23y = x2 , x = 0 (x ≥ 0).

6.12.y = sin x, y = cos x, x = 0, (x ≥ 0).

6.13.y = 20 − x2 , y = −8x.

6.14.y = 18− x2 , y = 32 − 18 − x2 .

6.15.y = 32 − x2 , y = −4x.

6.16.y = 2x, y = 5ex , y = 2, y = 5.

6.17. x2 + y2 = 36, 32y = x2 ( y ≥ 0).

6.18.y = 3x, y = 3x, x = 4.

6.19.y = 6 − 36 − x2 , y = 36 − x2 , x = 0 (x ≥ 0).

6.20.y = 254 − x2 , y = x − 52.

6.21.y = x, y =1x, x =16.

6.22.y = 2x, y = 7ex , y = 2, y = 7.

6.23.x = 27 − y2 , x = −6y.

6.24.x = 72 − y2 , 6x = y2 , y = 0 (y ≥ 0).

6.25.y = 6 − x2 , y = 6 − 6 − x2 .

6.26.y = 3 x, y = 3 , x = 4.

22x

6.27.y = sin x, y = cos x, x = 0, (x ≤ 0).

6.28. y =	1	, y = 6ex , y =1, y = 6.
	x

6.29.y = 3x, y = 3x, x = 9.

6.30.y =11− x2 , y = −10x.

6.31.x2 + y2 =12, x6 = y2 (x ≥ 0).

Задача 7. Найти площадь фигуры, ограниченной данными линиями.

y2 − 2y + x2 = 0,

7.1.y2 − 4y + x2 = 0, y = x3, y = 3x.

y2 − 6y + x2 = 0,

7.3.y2 − 8y + x2 = 0,

y = x3, y = 3x.

x2 − 4x + y2 = 0,

7.2.x2 − 8x + y2 = 0, y = 0, y = x3.

x2 − 2x + y2 = 0,

7.4.x2 − 4x + y2 = 0, y = 0, y = x.

y2 − 8y + x2 = 0,

7.5.y2 −10y + x2 = 0, y = x3, y = 3x.

y2 − 4y + x2 = 0,

7.7.y2 − 6y + x2 = 0, y = x, x = 0.

y2 − 6y + x2 = 0,

7.9.y2 −10y + x2 = 0, y = x, x = 0.

y2 − 2y + x2 = 0,

7.11.y2 − 4y + x2 = 0, y = 3x, x = 0.

y2 − 4y + x2 = 0,

7.13.y2 − 6y + x2 = 0, y = 3x, x = 0.

y2 − 2y + x2 = 0,

7.15.y2 − 6y + x2 = 0, y = x3, y = 0.

y2 − 2y + x2 = 0,

7.17.y2 −10y + x2 = 0, y = x3, y = 3x.

y2 − 4y + x2 = 0,

7.19.y2 −10y + x2 = 0, y = x3, y = 3x.

x2 − 4x + y2 = 0,

7.6.x2 − 8x + y2 = 0, y = 0, y = x.

x2 − 2x + y2 = 0,

7.8.x2 −10x + y2 = 0, y = 0, y = 3x.

x2 − 2x + y2 = 0,

7.10.x2 − 4x + y2 = 0,

y = x3, y = 3x.

x2 − 2x + y2 = 0,

7.12.x2 − 6x + y2 = 0,

y = x3, y = 3x.

x2 − 2x + y2 = 0,

7.14.x2 − 8x + y2 = 0,

y = x3, y = 3x.

x2 − 2x + y2 = 0,

7.16.x2 − 4x + y2 = 0, y = 0, y = x3.

x2 − 2x + y2 = 0,

7.18.x2 − 6x + y2 = 0, y = 0, y = x3.

x2 − 2x + y2 = 0,

7.20.x2 − 6x + y2 = 0, y = 0, y = x.

y2 − 2y + x2 = 0,

7.21.y2 − 4y + x2 = 0, y = x, x = 0.

y2 − 6y + x2 = 0,

7.23.y2 − 8y + x2 = 0, y = x, x = 0.

y2 − 4y + x2 = 0,

7.25.y2 − 8y + x2 = 0, y = x, x = 0.

y2 − 4y + x2 = 0,

7.27.y2 − 8y + x2 = 0, y = 3x, x = 0.

y2 − 2y + x2 = 0,

7.29.y2 −10y + x2 = 0, y = x3, x = 0.

y2 − 4y + x2 = 0,

7.31.y2 − 8y + x2 = 0, y = x3, x = 0.

Задача 8. Пластинка

x2 − 2x + y2 = 0,

7.22.x2 − 4x + y2 = 0, y = 0, y = 3x.

x2 − 4x + y2 = 0,

7.24.x2 − 8x + y2 = 0, y = 0, y = 3x.

x2 − 4x + y2 = 0,

7.26.x2 − 8x + y2 = 0,

y = x3, y = 3x.

x2 − 4x + y2 = 0,

7.28.x2 − 6x + y2 = 0,

y = x3, y = 3x.

x2 − 6x + y2 = 0,

7.30.x2 −10x + y2 = 0, y = x3, y = 3x.

D задана ограничивающими ее кривыми, - поверхностная

плотность. Найти массу пластинки.

D : x =1, y = 0, y2 = 4x	(y ≥ 0);	D : x2 + y2 =1, x2 + y2 = 4,
D : x =1, y = 0, y2 = 4x	(y ≥ 0);	x = 0, y = 0 (x ≥ 0, y ≥ 0);
8.1.	8.2.	x = 0, y = 0 (x ≥ 0, y ≥ 0);
= 7x2 + y.		= (x + y) (x2 + y2 ).
		= (x + y) (x2 + y2 ).

D : x =1, y = 0, y2 = 4x	D : x2 + y2 = 9,		x2 + y2 =16,
D : x =1, y = 0, y2 = 4x	( y ≥ 0);	x = 0, y = 0 (x ≥ 0, y ≥ 0);
8.3.	8.4.	x = 0, y = 0 (x ≥ 0, y ≥ 0);
= 7x2 2 + 5y.		= (2x + 5y) (x2 + y2 ).
		= (2x + 5y) (x2 + y2 ).
D : x = 2, y = 0, y2 = 2x	D : x2 + y2 =1,		x2 + y2 =16,
D : x = 2, y = 0, y2 = 2x	(y ≥ 0);	x = 0, y = 0 (x ≥ 0, y ≥ 0);
8.5.	8.6.	x = 0, y = 0 (x ≥ 0, y ≥ 0);
= 7x2 8+ 2y.		= (x + y) (x2 + y2 ).
		= (x + y) (x2 + y2 ).
D : x = 2, y = 0, y2 = x 2	D : x2 + y2 = 4,		x2 + y2 = 25,
D : x = 2, y = 0, y2 = x 2	(y ≥ 0);	x = 0, y = 0 (x ≥ 0, y ≤ 0);
8.7.	8.8.	x = 0, y = 0 (x ≥ 0, y ≤ 0);
= 7x2 2 + 6y.		= (2x − 3y) (x2 + y2 ).
		= (2x − 3y) (x2 + y2 ).
D : x =1, y = 0, y2 = 4x	(y ≥ 0);	D : x2 + y2 =1, x2 + y2 = 9,
D : x =1, y = 0, y2 = 4x	(y ≥ 0);	x = 0, y = 0	(x ≥ 0, y ≤ 0);
8.9.	8.10.	x = 0, y = 0	(x ≥ 0, y ≤ 0);
= x + 3y2.		= (x − y) (x2 + y2 ).
		= (x − y) (x2 + y2 ).
D : x =1, y = 0, y2 = x	(y ≥ 0);	D : x2 + y2 = 9,	x2 + y2 = 25,
D : x =1, y = 0, y2 = x	(y ≥ 0);	x = 0, y = 0	(x ≤ 0, y ≥ 0);
8.11.	8.12.	x = 0, y = 0	(x ≤ 0, y ≥ 0);
= 3x + 6y2.		= (2y − x) (x2 + y2 ).
		= (2y − x) (x2 + y2 ).