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# Кузнецов_математика

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∫∫∫y2ch(xy) dx dy dz;

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

Задача 5. Вычислить.

∫∫∫x2 z ch(xyz) dx dy dz;

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

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∫∫∫x dx dy dz;

V

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

∫∫∫15(y2 + z2 ) dx dy dz;

V

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

∫∫∫(1+ 2x3 ) dx dy dz;

V

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

z = xy, z = 0.

∫∫∫y dx dy dz;

V

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

∫∫∫(3x2 + y2 ) dx dy dz;

V

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

∫∫∫(4 + 8z3 ) dx dy dz;

V

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

z = xy, z = 0.

 ∫∫∫V dx dy dz ; + x + y + z 4 1 3 4 8 5.2. V : 1+ x + y + z =1, 3 4 8

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

∫∫∫(3x + 4y) dx dy dz;

V

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

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

∫∫∫(27 + 54y3 ) dx dy dz;

 V 5.6. V : y = x, y = 0, x =1, z = xy, z = 0. ∫∫∫V dx dy dz ; + x + y + z 5 1 16 8 3 5.8. V : x + y + z =1, 16 5 3

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

∫∫∫(15x + 30z) dx dy dz;

V

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

∫∫∫(1+ 2x3 ) dx dy dz;

V

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

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∫∫∫21xz dx dy dz;

V

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

∫∫∫(x2 + 3y2 ) dx dy dz;

V

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

 ∫∫∫ 10 x + 5 dx dy dz; 3 3

V

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

∫∫∫3y2 dx dy dz;

V

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

∫∫∫x2 dx dy dz;

V

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

∫∫∫63(1+ 2y ) dx dy dz;

V

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

∫∫∫

V

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;

V

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

z = x2 + y2 , z = 0.

∫∫∫(9 +18z) dx dy dz;

V

5.18. V :

∫∫∫

V

5.20. V :

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

 dx dy dz ; + x + y + z 6 1 2 4 6 x + y + z =1, 2 6 4

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

∫∫∫(8y +12z) dx dy dz;

V

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

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

∫∫∫(x + y) dx dy dz;

V

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

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

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 ∫∫∫V dx dy dz ; + x + y + z 6 1 6 4 16 5.25. V : x + y + z =1, 6 4 16

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

∫∫∫y2 dx dy dz;

V

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

∫∫∫(x2 + 4y2 ) dx dy dz;

V

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

∫∫∫x2 z dx dy dz;

V

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

∫∫∫xyz dx dy dz;

V

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 ; + x + y + z 6 1 8 3 5 5.30. V : x + y + z =1, 8 5 3

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.

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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 = 18x2 , 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.

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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 =11x2 , 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.

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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.

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

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

D : x = 2, y = 0, y2 = x 2 (y 0);

8.13.

= 2x + 3y2.

D : x = 1 , y = 0, y2 = 8x (y 0);

8.15.2

= 7x + 3y2.

D : x =1, y = 0, y2 = 4x (y 0);

8.17.

= 7x2 + 2y.

D : x2 + y2 = 4, x2 + y2 =16,

8.14.x = 0, y = 0 (x 0, y 0);

= (2y 3x)(x2 + y2 ).

D : x2 + y2 = 9, x2 + y2 =16,

8.16.x = 0, y = 0 (x 0, y 0);

= (2y 5x)(x2 + y2 ).

D : x2 + y2 =1, x2 + y2 =16,

8.18.x = 0, y = 0 (x 0, y 0);

= (x + 3y)(x2 + y2 ).

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D : x = 2, y2 = 2x, y = 0 ( y 0);

8.19.

= 7x24 + y2.

D : x = 2, y = 0, y2 = 2x (y 0);

8.21.

= 7x24 + y.

D : x = 2, y = 0, y2 = x 2 (y 0);

8.23.

= 7x22 + 8y.

D : x =1, y = 0, y2 = 4x (y 0);

8.25.

= 6x + 3y2.

D : x = 2, y = 0, y2 = x 2 (y 0);

8.27.

= 4x + 6y2.

D : x = 1 , y = 0, y2 = 2x (y 0);

8.29.2

= 4x + 9y2.

D: x = 1 , y = 0, y2 =16x (y 0);

8.31.4

=16x + 9y2 2.

D : x2 + y2 =1, x2 + y2 = 4,

8.20.x = 0, y = 0 (x 0, y 0);

= (x + 2y)(x2 + y2 ).

D : x2 + y2 =1, x2 + y2 = 9,

8.22.x = 0, y = 0 (x 0, y 0);

= (2x y)(x2 + y2 ).

D : x2 + y2 =1, x2 + y2 = 25,

8.24.x = 0, y = 0 (x 0, y 0);

= (x 4y)(x2 + y2 ).

D : x2 + y2 = 4, x2 + y2 =16,

8.26.x = 0, y = 0 (x 0, y 0);

= (3x y)(x2 + y2 ).

D : x2 + y2 = 4, x2 + y2 = 9,

8.28.x = 0, y = 0 (x 0, y 0);

= (y 4x)(x2 + y2 ).

D : x2 + y2 = 4, x2 + y2 = 9,

8.30.x = 0, y = 0 (x 0, y 0);

= (y 2x)(x2 + y2 ).

Задача 9. Пластинка D задана неравенствами, - поверхностная плотность. Найти массу пластинки.

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