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

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

−

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

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

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