функции многих переменных идз

1:

f(x; y) =

8 ln(1x + y2

;

x + y2 6= 0 ;

<

+ x + y)

1

;

x + y2 = 0

2: f(x; y) =

:x2y2

;

sin(x2y)

3: f(x; y) =

2x + y

;

px1

+cos y

2

2

8

2

2y2

4: f(x; y) =

+ y2

; x + y 6= 0 ;

x2

<

a

; x2 + y2 = 0

5: f(x; y) =

:

x2y

;

1

cos(xpy)

8

y2

6: f(x; y) =

<

x2 + y2

; (x; y) 6= (0; 0) ;

a

; x = 0 = y

7: f(x; y) =

: xy2

;

sin(xy2)

sin x

8:

f(x; y) =

;

2

y2

px1

+cos x

2

2

9: f(x; y) =

8

+ y2

; x

+ y

6= 0 ;

x2

<

a

; x = 0 = y

10:

f(x; y) =:sin2 x sin y

;

x2 + y2

8

x3

y3

11: f(x; y) =

; x 6= y ;

x

y

<

a

; x = y

12: f(x; y) =

: xy

;

sin(xy + y

2

)

8

x sin y

; (x; y) 6= (0; 0) ;

13: f(x; y) =

x2

+ y2

<

a

; x = 0 = y

ln(1 + xy)

14:

f(x; y) =

:

;

x2 + y2

1p cos(xy)

15:

f(x; y) =

;

2x2

+ y2

8

x2

y2

16: f(x; y) =

; x 6= y ;

x

y

<

a

; x = y

17:

f(x; y) =

: y sin x

;

x2 + 2y2

18: f(x; y) =

px + y

;

p

;

19:

f(x; y) =

x2 y2 ln(1 + x + y)

x2 + y

21

; (x; y) 6= (0; 0)

; x = 0 = y

20:

21:

22:

23:

24:

25:

x2y

f(x; y) = sin(x2y);

f(x; y) = x(1 cos y);

			x2 + y2
	pln(1 + xy)
8
<
f(x; y) =				x2 + y2
:				1
f(x; y) =	x + 2y					;
	p
		2x2 + y2
8				x2
f(x; y) =				2 + y2
< x				a
:

f(x; y) =	xy2
	1 cos(yp		)	:
		x

;	x2 + y2	6= 0	;
;	x2 + y2	= 0

;

ИДЗ-4. Дифференцирование функции нескольких переменных, заданной явно

Найти первые и вторые частные производные функции z(x; y); заданной явно. Убедиться в том, что zxy00 = zyx00 : Записать выражения для первого и второго дифференциалов функции z(x; y):

1:

z = ex2 y2 :

3:

z = tg(y=x):

5:

z = sin(x2 y):

7:

z = arcsin(x y):

9:

z =

arcctg(x

3y):

2

2

:

11:

z = e2x

+y

13:

z = tg p

xy:

17:

z =

p

15:

z = sin

x3y:

arccos(4x

y):

19:

z = arctg(2x y):

21:

p

x+y

:

z = e

23:

z = arccos(x 5y):

25:

z = cos(3x2 y3):

2:	z = ctg(x + y)
4:	z = cos(xy2):
6:	z = arctg(x + y):
8:	z = arccos(2x + y):
10:	z = ln(3x2 2y2):
12:	z = ctg(y=x):
14:	z = cos(x2y2 5):
16:	z = arcsin(x 2y):
18:	z = arctg(5x + 2y):
20:	z = ln(4x2 5y3):
22:	z = arcsin(4x + y):
24:	z = sin p
		yx:

ИДЗ-5. Дифференцирование функции нескольких переменных, заданной неявно

Вычислить значения первых частных производных функции z(x; y); заданной неявно, в данной точке M0(x0; y0; z0) с точностью до двух знаков после запятой.

1:	x3 + y3	+ z3	3xyz = 4;	M0	(2; 1; 1):
2:	x2 + y2	+ z2	xyz = 2;	M0	( 1; 01; 1):
3:	3x 2y + z = xy + 5;			M0	(2; 1; 1):
4:	ez + x + 2y = 4;			M0	(1; 1; 0):

22

5:	x2 + y2 + z2 z 4 = 0;			M0(1; 1; 1):
6:	x3 + 3xyz + 3y = 7;			M0(1; 1; 1):
7:	cos2 x + cos2 y + cos2 z = 23 ;			M0( 4 ;		3	; 4 ):
						4
8:	ez 1 = cos x cos y + 1;			M0(0; ; 1):
	x2 + y2 + z2 6x = 0;				2
9:				M0(1; 2; 1):
10:	xy = z2 1;			M0(0; 1; 1):
11:	x2 2y2 + 3z2 yz + y = 2;			M0(1; 1; 1):
12:	x2 + y2 + z2 + 2xz = 5;			M0(0; 21; 1):
13:	x cos y + y cos z + z cos x = =2;			M0(2; =2; ):
14:	3x2y2 + 2xyz2 2x3z + 4y3z = 4;			M0(2; 1; 2):
15:	x2 2y2 + z2 4x + 2z + 2 = 0;			M0(1; 1; 1):
16:	x + y + z + 2 = xyz;			M0(2; 1; 1):
17:	x2 + y2 + z2 2xz = 2;			M0(0; 1; 1):
18:	ez xyz x + 1 = 0;			M0(2; 1; 0):
19:	x3 + y3 + z3 3xyz = 4;			M0(2; 1; 1):				M0(0; 2; 2):
20:	x2 2xy 3y2 + 6x 2y + z2 8z + 20 = 0;
21:	x2 + y2 + z2 = y z + 3;							M0(1; 2; 0):
22:	x2 + y2 + z2 + 2xy yz 4x 3y z = 0;							M0(1; 1; 1):
23:	x2 y2 z2 + 6z + 2x 4y + 12 = 0;							M0(0; 1; 1):
25:	p			0
24:	x2 + y2 + z2 3z = 3;			M0	(4; 3; 1):
	x2 + 2y2 + 3z2 = 59;			M (3; 1; 4):

ИДЗ-6. Уравнения для функции нескольких переменных и ее производных

Проверить, удовлетворяет ли указанному уравнению данная функция нескольких переменных.

1:

2:

3:

4:

5:

6:

7:

8:

9:

10:

11:

12:

13:

14:

15:

16:

x2

@2u

+ 2xy

@2u

+ y2

@2u

= 0;

u = y

:

2

@x@y

2

@x

@y

x

@u

@u

3

x

3

3

x2@x

+ y

@y2

= 3(x63 y

);

u = ln y

+ x

y

:

@ u

+ y2

@

u

= 0;

u = ln(x2 + (y + 1)2):

2

2

@x

@y

y

@2u

= (1 + y ln x)@u

;

u = xy:

@x@y

@x

xy

x

@u

+ y

@u

= 2u;

u =

:

x+y

@x

@y

x2

@2u

+ y2

@2u

= 0;

u = exy:

2

2

@x

@y

a2

@2u

=

@2u

= 0;

u = sin2(x ay):

@x2

@y2

@2u

@2u

@

2u

q

1

x2

@2u

y2

@2u

= 0;

u = y

xy :

@x2

@y2

@x2 + @y2 + 2 @z2 = 0;

u =

p

:

x2+y2+z2

a2

@2u

= y2

@2u

= 0;

u = e( cos(x+ay):

2

2

@x

@y

u = (x y)(y z)(z x):

@u@x + @u@y + @u@z = 0;

x

@u

+ y

@u

= u;

u = x ln y :

@x

@y

x

y

@u

x@u

= 0;

ln(x2 + y2):

@x

@y

u =

2

+ arcsin(xy):

x2 @u

xy @u + y2 = 0;

u = y

@x

@y

3x

2 @

2

@

2

u

2

u

xy

u

2 @

x

2xy

+ y

+ 2xy = 0;

u = 0;

u = e

:

2

@x2

@x@y

@y2

@ u

= 0;

u = arctg

x+y

:

@x@y

1 xy

23

17:

@2u

+ y2

@2u

= 0;

u = ln(x2 + y2 + 2x + 1);

2

2

@x

@y

2x+3y

18: x

@u

+ y

@u

+ u = 0;

@x

@y

u = x2+y2

20: x

@u

+ y

= 2u;

u = (x2 + y2) tg x :

@u

2

@u

2

@u

2

19:

@x

+

@y

= 1;

p

y

@x

@y

+ @z

u = x2 + y2 + z2:

@u

21:

9

@2u

+

@2u

= 0;

u = e (x+3y) sin(x + 3y):

2

2

@x

@y

22:

x2

@2u

+ 2xy

@2u

+ y2

@2u

= 0;

u = xey=x:

@x2

@x@y

@y2

23:

@2u

+ y2

@2u

= 0;

u = arctg y

:

2

2

@x

@y

x

24:

x

@u

+ y

@u

= 0;

u = arctg x

:

@x

@y

y

25:

@u @2u

@u @2u

= 0;

u = ln(x + e y):

@x

@y @x2

@x@y

0.1ИДЗ-7. Исследование функции нескольких переменных на локальный экстремум

Исследовать на локальный экстремум функцию нескольких переменных.

1: z = ypx 2y2 x + 14:

2: z = x3 + 8y3 6xy + 5:

3: z = 1 + 15x 2x2 xy 2y2:

4: z = 1 + 6x x2 xy y2:

5: z = x3 + y2 6xy + 39x + 18y + 20: 6: z = 2x3 + 2y3 6xy + 5:

7: z = 3x3 + 3y3 9xy + 10:

8: z = x2 + xy + y2 + x y + 1: 9: z = 4(x y) x2 y2:

10: z = 6(x y) 3x2 3y2:

11: z = x2 + xy + y2 6x 9y:

12: z = (x 2)2 + 2y2 10: 13: z = (x 5)2 + y2 + 1: 14: z = x3 + 8y3 6xy + 5:

15: z = 2xy 2x2 4y2:

16: z = xpy x2 y + 6x + 3:

17: z = 2xy 5x2 3y2 + 2: 18: z = xy(12 x y):

19: z = xy x2 y2 + 9:

20: z = 2xy 3x2 2y2 + 10:

21: z = xp3 + 8y3 6xy + 1:

22: z = y x y2 x + 6y:

23: z = x2 xy + y2 + 8x 6y + 20:

24: z = xy(6 x y):

25: z = x2 + y2 xy + x + y:

ИДЗ-8. Наибольшее и наименьшее значение функции нескольких переменных в замкнутой области

Найти наибольшее и наименьшее значения функции нескольких переменных в замкнутой области.

24

1:	z = 3x + y xy;		D : y = x; y = 4; x = 0:
2:	z = xy x 2y;		D	: y = x; y = 0; x = 3:
3:	z = x2 + 2xy 4x + 8y;			: y = 0; x = 1; x = 0; y = 2:
			D
4:	z = 5x2 3xy + y2;			: y = 0; y = 1; x = 1:
			D
5:	z = x2 y2 + 2xy 4x;			: x y + 1 = 0; y = 0;			x = 3:
			D
6:	z = x2 + y2 2x 2y + 8;			: x + y 1 = 0; y = 0;			x = 0:
			D
7:	z = 2x3 + xy2 + y2;			: y = 6; y = 0; x = 0;			x = 1:
			D
8:	z = 3x + 6y x2 xy y2;			: y = 1; y = 0; x = 0;			x = 1:
			D
9:	z = x2 2y2 + 4xy 6x 1;			: y + x 3 = 0; y = 0;			x = 0:
			D
10:	z = x2 + 2xy 10;			: y = 0; y = x2 4:
			D
11:	z = xy 2x y;			: y = 0; y = 4; x = 0;			x = 3:
			D
12:	z = 21 x2 xy;			: y = 8; y = 2x2:
			D
13:	z = 3x2 + 3y2 2x 2y + 2;			: y + x 1 = 0; y = 0;			x = 0:
			D
			D : y + q
14:	z = 2x2 + 3y2 + 1;		D	: y = 9 49 x2; y = 0:
15:	z = x2	2xy y2 + 4x + 1;		x + 1 = 0; y = 0;			x = 3:
16:	z = 3x2 + 3y2 x y + 1;			: x y 1 = 0; y = 0;			x = 5:
			D
17:	z = 2x2 + 2xy 21 y2 4x;			: y = 2x; y = 2; x = 0:
			D
18:	z = x2 2xy + 25 y2 2x;			: y = 0; y = 2; x = 0;			x = 2:
			D
19:	z = xy 3x 2y;			: y = 0; y = 4; x = 0;			x = 4:
			D
20:	z = x2 + xy 2;			: y = 4x2 4; y = 0:
			D
21:	z = x2y(4 x y);			: y = 6 x; y = 0; x = 0:
			D
22:	z = x3 + y3 3xy;			: y = 1; y = 2; x = 0; x = 2:
			D
23:	z = 4(x y) x2 y2;			: 2y + x = 4; x 2y = 4; x = 0:
			D
24:	z = x2 + 2xy y2 4x;			: y = x + 1; y = 0; x = 3:
			D
25:	z = 6xy 9x2 4x;			: y = 0; y = 2; x = 0; x = 1:
			D

ИДЗ-9. Изменение порядка интегрирования в кратных интегралах

Изобразить область интегрирования; изменить порядок интегрирования.

1

0

0

0

1:

R2 dy pR2+y

f dx + R1 dy pR

f dx:

y

p

1

0

2

0

2:

R

dy

R

f dx +

R

dy

pR2 y

f dx:

0

p

y

1

p

2

p

y

2 y2

1

2

3:

R1

dy

Rpy

R2

p2R0

y

f dx:

f dx +

dy

0

0

1

R0

1

R0

f

0

R1

0

4:

dy

R0 0

dy

dx +

f dx:

2R x

R

R

5:

R 2

f dy +

p

dx

p

2

1 dx x f dy:

1=p

2

arcsin y

1

6:

R0

dy

R0

f dx +

1=Rp

dy R0

f dx:

2

1

p

p

2+y

0

y

7:

R2 dy

R0

f dx + R1 dy

R0

f dx:

25

1

0

2

e

ln y

2

R

Rp

R

R

8:

0

dy

p

f dx +

1

dy

1

f dx:

y

1

2 x

0

x

9:

R 2

R

f dy +

R

R

p

0

1 dx

0

p

3

1

0

e

0

0

1R

R

1R

R

10:

2

dx

p

f dy +

p

dxp

f dy:

4

x2

3

4

x2

R

R

11:

dx

pRy

f dy +

dx

R

f dy:

0

1 x

2

1

ln x

1

3

2

2 y

Rsin y

cos y

12:

R=4

R

=2 R

dy

f dx +

dy

f dx:

0

0

1

0

13:

R01

dy

R0

0 f dx + R=4

0dy

R0

0 f dx:

R

R

Rx

14:

R

dx

f dy:

2

(2+x)

1

p

p

3

15:

1

y

e

1

0

R01 dy

R0 0 f dx + R1 2

lnRy

R

R

dy

f dx:

16:

f dx +

R

R2 y

0

p

y

1

p

p

1

0

2

0

17:

R0

dy

Ry

f dx + R1

dy p

R2 y2

f dx:

1

y2

0

2

2 y

0

18:

Rp3

R0

R1

R0

2

dy

f dx +

dy

f dx:

0

19:

R0

dxp

4 Rx2

2 f dy + pR

dx p

4R x2

f dy:

3

1

y

0

e

0

0

20:

R1

R

R1

dy

R

f dx:

2

(2+y)

1

py

21:

R

dy

R

2

Rp

lnRp

3

2

f dx +

dy

f dx:

0

0

1

y

1

x

R

2

=2

2 x

22:

R=4

Rsin x

R cos x

dx

f dy +

dx

f dy:

0

0

R0

1

0

R0

23:

R0 1dx

R=40dx

0

f dy +

f dy:

R

pR2 y

R

24:

R 2

f dx +

p

2

dy y

1

x2

2

2 x

25:

R0

dx

R0

f dy + R1

dx

R0

f dy:

ИДЗ-10. Вычисление двойных интегралов

Вычислить двойные интегралы в указанных областях.

26

1.

(12x2y2

+ 16x3y3)dxdy;

D : x = 1; y = x2; y = px:

RR

D

D : x = 1; y = x2; y = px:

2.

(9x2y2

+ 48x3y3)dxdy;

RR

D

96x3y3)dxdy;

D : x = 1; y = x3; y = p3 x:

3.

(36x2y2

RR

D

D : x = 1; y = x3; y = p3 x:

4.

(18x2y2

+ 32x3y3)dxdy;

RR

D

D : x = 1; y = x2; y = p3 x:

5.

(27x2y2

+ 48x3y3)dxdy;

RR

D

D : x = 1; y = x2; y = p3 x:

6.

(18x2y2

+ 32x3y3)dxdy;

RR

D

D : x = 1; y = x3; y = px:

7.

(18x2y2

+ 32x3y3)dxdy;

RR

D

D : x = 1; y = x3; y = px:

8.

(27x2y2

+ 48x3y3)dxdy;

RR

D

D : x = 1; y = x2; y = px:

9.

(4xy + 3x2y2)dxdy;

RR

D

D : x = 1; y = x2; y = px:

10.

(9x2y2

+ 12xy)dxdy;

RR

D

D : x = 1; y = x3; y = p3 x:

11.

(9x2y2

+ 8xy)dxdy;

RR

D

D : x = 1; y = x3; y = p3 x:

12.

(18x2y2

+ 24xy)dxdy;

RR

D

D : x = 1; y = x2; y = p3 x:

13.

(27x2y2

+ 12xy)dxdy;

RR

D

D : x = 1; y = x2; y = p3 x:

14.

(18x2y2

+ 8xy)dxdy;

RR

D

54 xy +

119 x2y2)dxdy;

D : x = 1; y = x3; y = px:

15.

(

RR

D

54 xy + 9x2y2)dxdy;

D : x = 1; y = x3; y = px:

16.

(

RR

D

D : x = 1; y = x2; y = px:

17.

(24xy 48x3y3)dxdy;

RR

D

D : x = 1; y = x2; y = px:

18.

(6xy + 24x3y3)dxdy;

RR

D

D : x = 1; y = x3; y = p3 x:

19.

(4xy + 16x3y3)dxdy;

RR

D

D : x = 1; y = x3; y = p3 x:

20.

(4xy + 16x3y3)dxdy;

RR

D

D : x = 1; y = x2; y = p3 x:

21.

(44xy + 16x3y3)dxdy;

RR

D

D : x = 1; y = x2; y = p3 x:

22.

(4xy + 176x3y3)dxdy;

RR

D

27

23. RR (xy 4x3y3)dxdy; D : x = 1; y = x3; y = px:

	D		D : x = 1; y = x2; y = px:
24.	(4xy + 176x3y3)dxdy;
	RR
	D	+ 253 x4y4)dxdy;	D : x = 1; y = x2; y = px:
25.	(6x2y2
	RR

D

ИДЗ-11. Вычисление площадей фигур

Вычислить площади фигур, ограниченных данными линиями.

3:

x + y = 72; 6y = x (y 6 0):

4:

x = p

p

1:

y = x3

; y = 4ex; y = 3; y = 4:

2:

x =

36 y2

; x = 6

36 y2

:

2

2

2

8

y2; x =

2y:

p

5:

y =

3

; y = 8ex; y = 3; y = 8:

6:

y =

x

; y =

1

; x = 16:

x

2

2x

7:

x = 5 y2; x = 4y:

8:

x2 + y2 = 12; p

= x2(y 6 0):

6y

p

11:

y = p24 x ; 2 3y = p

>

x

1

>

:

10:

y = 2 ; y =

2x ; x = 16:

9:

x =

p

36 y2; x = 6

36 y2

2

p

x2;

x = 0(x 0):

12:

y = sin x;

y = cos x; x = 0(x 0):

13:

y = 20 x2; y = 8x:

14:

y = p

; y = p

p

:

18 x2

2

18 x2

15:

y = 32 x2; y = 4x:

16:

y = x2

; y = 5ex; y = 2; y = 5:

17:

x2 + y2 = 36; 3p

= x2(y 6 0):

18:

y = 3p

x; y = x3 ; x = 4:

2y

21:

y = x; y = x ; xp

x

19:

y =

23 p

36 y2

:

20:

y = 254 = x2; y = x 25 :

x; x = 6

p

1

= 16:

22:

y = 2 ; y = 7ex; y = 2; y = 7:

25:

y = p6 x2; y = p6

p6 x2:

p

>

24:

; 6x = y2; y = 0(y

0):

23:

x = 27 y2

; x = 6y:

x = 72

y2

0.2ИДЗ-12. Вычисление тройных интегралов

Вычислить тройные интегралы в указанных областях.

1a: 2y2exydxdydz;
RRRx = 0;	y = 1;	y = x;
V
V z = 0;	z = 1:
RRRx = 2;	y = ;	z = 1;
2a: x2z sin(xyz)dxdydz;
V

V

x = 0; y = 0; z = 0:

3a: RRR 2y2 cos 2xydxdydz;

	RRR	= 10x;				y = 0;		x = 1;
1b:		xdxdydz;
	V
V yz = xy;						z = 0;		:
	RRR			dxdydz			;
		+ +				= 1
2b: V			(1+ x3 + y4 + z8 )4
V	x		y		z
	x = 0; y = 0; z = 0:
	3	4			8

3b: RRR 15(y2 + z2)dxdydz;

V		z = 2:	V
z = 0;			x = 0; y = 0 z = 0:
V x = 0;		y = 2; y = 4x;	V z = x + y; x + y = 1;
4a: RRR	8y2ze2xyzdxdydz;		4b: RRR (3x + 4y)dxdydz;

	V
	z = 0; y = 0; z = 0:
V	x = 1; y = 2; z = 1;

V	V	+ y2); z = 0:
	z = 5(x2
	x = 1; y = 0; y = x;

28

5a: RRR x2 sin(3xy)dxdydz;

	V	y = 2x;				z = 1;
	x = 1;
V z = 0;		y 0;				z = 0:
6a:	y2 cos xyzdxdydz;
	V	y = 2x;				y = 0;
RRRx = 1;
V z = 0;		z = 36:
7a:	y2 cos	xy		dxdydz;
	V	4
	x = 0;	y =		1;
RRR						y = x=2;
V
z = 0;		z = 2:
8a:	x2z sin xyz dxdydz;
	V	4
		y = 2 ;				z = 4;
RRRx = 1;
V x = 0; y = 0; z = 0:
9a:	y2e xydxdydz;
	V
RRR
V	x = 0;	y = 2; y = 4x;
z = 0;		z = 1:
10a:	2y2exyzdxdydz;
	V	y = 1;			z = 1;
	RRRx = 1;
V x = 0;		y = 0; z = 0:
11a:	y2 cos 2xydxdydz;
	V	y = 2;				y = 2x;
	RRRx = 0;
V z = 0;		z = 1:
12a:	x2z sin(xyz)dxdydz;
	V	y = 1;			z = 1;
	RRRx = 2;
V x = 0;		y = 0; z = 0:
13a:	8y2ze xyzdxdydz;
	V
	RRR
V	x = 2;	y = 1; z = 2;
z = 0;		y = 0;				z = 0:
14a:	y2z cos xyz dxdydz;
	V	3
		y = 1 ;				z = 2 ;
	RRRx = 3;

V

x = 0; y = 0; z = 0:

RRRy = 9x;							y = 0;		x = 1;
5b:		(1 + 2x3)dxdydz;
V
V z = p							z = 0:
				xy;
RRR			= x;				y = 0;		x = 1;
6b:		(27 + 54y3)dxdydz;
V
V z =yp							z = 0:
				xy;
RRRx = 1;							y = 0;		x = 1;
7b:		ydxdydz;
V
V z = xy;							z = 0:
RRRx			dxdydz
			+ y		+ z = 1;
8b:			(1+ x3		+ y4 + z8 )5 ;
V
V		= 0; y = 0; z = 0:
	x
16		8					3
9b:		(3x2 + y2)dxdydz;
V
RRR

V

z = 10y x + y = 1;

x = 0; y = 0; z = 0:

10b: RRR (15x2 + 30z)dxdydz;

V

V

z = x2 + 3y2 z = 0; x = 1; y = 0; y = x:

11b: RRR (4 + 8z3)dxdydz;

V

y = x y = o; x = 1;

Vz = pxy z = 0:

12b: RRR (1 + 2x3)dxdydz;

V

y = 36x y = o; x = 1;

V

z = p

z = 0:

xy

RRRy = 1x;

y = 0;

x = 2;

13:b

21xzdxdydz;

V

V z = xy;

z = 0;

:

RRR

dxdydz

;

+

+

= 1;

14b: V

(1+

x

+ y8 + z3 )6

10

x

y

z

V

x

= 0; y = 0; z = 0:

10

8

3

29

15a: RRR y2 cos 2xy dxdydz;

	V	z = 2 2		:
z = 0;
V	x = 0; y = 1; y = x ;
16a:	RRR
	2yx2z sin xyzdxdydz;
	V		y = 0;	z = 0:
x = 0;
V	x = 1;	y = 1 ; z = 1;
17a:	RRRx = 0;	y = 1;		y = 2x;
	y2 cos xydxdydz;
	V
V z = 0;		z = 2:
18a:	RRRx = 2;	y = 1=2; y = 1=2;
	2x2z sin 2xyz dxdydz;
	V
V z = 0;			y = 0;	z = 0:
19a:	RRR
	x2 sin 2xydxdydz;
	V		z = 8:
	z = 0;
V	x = 1; y = x; y = 0 ;
20a:	x2z sin xyz dxdydz;
	RRRx = 1;	y = 4;		z = ;
	V		2
V x = 0;		y = 0; z = 0:
21a:	RRR
	2y2 cos xydxdydz;
	V	z = 2:
z = 0;
V	x = 0;	y = 1; y = x;
22a:	RRRx = 1;	y = 1;		z = 1;
	y2z cos(xyz)dxdydz;
	V
V x = 0;		y = 0; z = 0:
23a:	RRRx = 2;	y	= x; y = 0;
	x2 sin		xy	dxdydz;
	V		2
V z = 0;		z = :
24a:	y2z cos xyz dxdydz;
	RRRx = 9;	y = 1;		z = 2 ;
	V		9
V x = 0;		y = 0; z = 0:
25a:	RRRx = 1;	y = 0;		y = 2x;
	x2 sin xydxdydz;
	V
V z = 0;		z = 4 :

15b: RRR 15(x2 + 3y2)dxdydz;

V	V
	x = 0; y = 0 z = 0:
	z = 10x; x + y = 1;

RRR

16b: (60y + 90z)dxdydz;

V

	V	x = 1; y = 0; y = x;
		z = x2 + y2; z = 0:

17b: RRR 103 x + 53 dxdydz;

V

y = 9x y = o; x = 1;

Vz = pxy; z = 0:

RRR

18b: (9 + 8z)dxdydz;

V

y = 4x y = o; x = 1;

Vz = pxy z = 0:

RRRy = 2x;

y = 0;

x = 2;

19b:

3y2dxdydz;

V

V z = xy;

z = 0;

:

RRR+

dxdydz

;

+

= 1;

20b: V

(1+ x2

+ y4

+ z6 )4

x

y

z

V x = 0; y = 0; z = 0:

2

4

6

RRRz = 10(x + 3y);

x + y = 1;

21b:

x2dxdydz;

V

V x = 0; y = 0; z = 0:

RRRx = 1;

y = 0;

y = x;

22b:

(8y + 12z)dxdydz;

V

V z = x2 + y2; z = 0:

RRR

= x;

y = 0; x = 1;

23b:

63(1 + 2p

y

)dxdydz;

V

V z =yp

z = 0:

xy;

RRRx = 1;

y = 0;

y = x;

24b:

(x + y)dxdydz;

V

V z = 30x2 + 60y2; z = 0:

RRR+

dxdydz

+

= 1;

25b: V

(1+ x6

+ y4 +

z

)4 ;

16

x

y

z

V x = 0; y = 0; z = 0:

6

4

16

30