Приложение определенных интегралов к решению задач геометрии и физики. Ляпунова М.Г. - 31 стр.

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Амурский Государственный Университет 31
1.12
0826
22
=+−++ yxyx
,
106
2
++= xxy
.
=
=
,sin2
,cos28
3
3
ty
tx
4
≥
x
.
,
2
sin
2
θ
ρ
=
2
0
π
θ
≤≤
.
1.13
0,63;4
2
=−== yxxyx
,
40
≤≤
x
.
=
=
,sin24
,cos2
ty
tx
4
≥
y
.
θρ
3sin
a=
.
1.14
0,932
2
=−+= yxxy
.
=
=
,sin2
,cos24
3
3
ty
tx
2
≥
x
.
,cos2
θρ
=
1
≥
ρ
.
1.150,56
2
=++= yxxy
.
−=
−=
),cos1(4
),sin(4
ty
ttx
π
80
≤≤
x
, 6
≥y
.
ϕρ
a
=
,
ϕρ
cos
a=
.
1.16
6116
23
−+−= xxxy
,
0
=
y
.
=
=
,sin4
,cos6
ty
tx
32
≥y
.
),cos1(
ϕρ
−=
a
πϕ
≤≤
0.
1.17
xyxy −=+=
4,)2(
2
.
−=
−=
),cos1(4
),sin(4
ty
ttx
6
≥y
,
π
20
≤≤
x
.
ϕρ
sin
a=
,
ϕρ
sin2
a=
.
1.18
.0,5,1,
2
==== yxxaxy
=
=
,sin3
,cos32
3
3
ty
tx
312
≥x
.
ϕρ
cos
a
=
,
ϕρ
cos2
a
=
.
1.19
.4,3
=+=
yxxy
=
=
,sin2
,cos6
ty
tx
3
≥y
.
a=
ρ
,
ϕρ
sin
a
=
.
1.20
+=
−
a
x
a
x
ee
a
y
2
,
.0,
== yax
+=
+=
.sin22
,cos22
ty
tx
a
=
ρ
,
ϕρ
cos
a=
.
1.21
.
4
,
4
8
2
2
x
y
x
y =
+
=
−=
−=
),cos1(2
),sin(2
ty
ttx
2
≥y
,
π
40
≤≤
x
.
ϕρ
cos
2
1
+=
.
1.22
.0,cos,1
==+= yxyxy
=
=
,sin2
,cos22
3
3
ty
tx
1
≥
x
.
ϕϕρ
sincos
−=
.
1.23
232
xxy −=
,
2
=
x
.
=
=
,sin
,cos16
3
3
ty
tx
36
≥x
.
ϕρ
sin3
=
,
ϕρ
sin5
=
.
1.24
2,
9
,4
2
2
=== y
x
yxy
.
−=
−=
),cos1(3
),sin(3
ty
ttx
3
≥y
,
π
60
≤≤
t
.
−=
2
cos2
π
ϕρ
.
1.25
,16,)4(
22
xyxy −=−=
0
=
y
.
−=
+=
,sin4cos5
,sin5cos4
tty
ttx
π
60
≤≤
t
.
ϕρ
sin
=
,
,
4
cos2
−=
π
ϕρ
πϕ
4
3
0
≤≤
.
Амурский Государственный Университет                                                   31

       x 2 +y 2 +6 x −2 y +8 =0 ,    x =8 2 cos 3 t ,                       θ         π
1.12        2                                   3
                                                       x ≥4 .     ρ =sin 2     , 0 ≤θ ≤ .
       y =x +6 x +10 .                y = 2 sin   t ,                       2         2
       x =4; y =3 x 2 −6 x, y =0 ,    x = 2 cos t ,
1.13                                                y ≥4 .       ρ =a sin 3θ .
       0 ≤x ≤4 .                     y =4 2 sin t ,
                                     x =4 2 cos 3 t ,
1.14   y =2 x 2 +3x −9, y =0 .                 3
                                                       x ≥2 .     ρ =2 cosθ , ρ ≥1 .
                                      y = 2 sin t ,
                                      x =4(t −sin t ),
                                                                 ρ =aϕ ,
1.15   y =x 2 +6 x +5, y =0 .        y =4(1 −cos t ),
                                                                  ρ =a cos ϕ .
                                     0 ≤x ≤8π , y ≥6 .
       y =x 3 −6 x 2 +11x −6 ,       x =6 cos t ,                ρ =a(1 −cos ϕ ),
1.16                                               y ≥2 3 .
                    y =0 .            y =4 sin t ,               0 ≤ϕ ≤π .
                                      x =4(t −sin t ),
                                                                 ρ =a sin ϕ ,
                                     y =4(1 −cos t ),
                      2
1.17   y =( x +2) , y =4 −x .
                                                                  ρ =2a sin ϕ .
                                     y ≥6 , 0 ≤x ≤2π .
                                     x =32 cos 3 t ,             ρ =a cosϕ ,
1.18            2
       xy =a , x =1, x =5, y =0.              3
                                                      x ≥12 3 .
                                      y =3 sin t ,               ρ =2a cosϕ .
                                     x =6 cos t ,                ρ =a ,
1.19   xy =3, x +y =4.                             y≥ 3.
                                      y =2 sin t ,               ρ =a sin ϕ .
         a a      − 
              x     x
       y=  e   +e  a 
                      ,             x =2 +2 cos t ,             ρ =a ,
1.20     2                        
                                      y =2 +2 sin t.             ρ =a cos ϕ .
         x =a, y =0.
                                      x =2(t −sin t ),
            8          x2                                           1
1.21   y=        , y =    .          y =2(1 −cos t ),            ρ = +cos ϕ .
          4 +x 2       4                                             2
                                     y ≥2 , 0 ≤x ≤4π .
                                     x =2 2 cos 3 t ,
1.22   y =x +1, y =cos x, y =0.                3
                                                       x ≥1 .     ρ =cos ϕ −sin ϕ .
                                      y = 2 sin t ,
                                     x =16 cos 3 t ,             ρ =3 sin ϕ ,
1.23    2
       y =x −x  3     2
                          , x =2 .                   x ≥6 3 .
                                              3
                                      y =sin t ,                 ρ =5 sin ϕ .
                                      x =3(t −sin t ),
                x2                                                         π
1.24            2
       y = x y = , y =2 .
          4  ,                       y =3(1 −cos t ),            ρ = 2 cosϕ − .
                9                                                             2
                                     y ≥3 , 0 ≤t ≤6π .
                                                                  ρ =sin ϕ ,
                                     x =4 cos t +5 sin t ,                 π
       y =( x −4) 2 , y =16 −x 2 ,                               ρ = 2 cosϕ − ,
1.25                                 y =5 cos t −4 sin t ,                   4
                     y =0 .
                                            0 ≤t ≤6π .                  3
                                                                  0 ≤ϕ ≤ π .
                                                                        4

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