Интегральное исчисление функции одной переменной. - 41 стр.

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§2. ïÐÒÅÄÅÌÅÎÎÙÊ ÉÎÔÅÇÒÁÌ, ÏÓÎÏ×ÎÙÅ Ó×ÏÊÓÔ×Á. . . 41
I.2. ëÒÉ×ÁÑ ÚÁÄÁÎÁ ÐÁÒÁÍÅÔÒÉÞÅÓËÉ: x = x(t), y = y(t), t ∈ [α, β], ÔÏÇÄÁ
ÄÌÉÎÁ ËÒÉ×ÏÊ ×ÙÞÉÓÌÑÅÔÓÑ ÐÏ ÆÏÒÍÕÌÅ:
l =
β
Z
α
p
[x
0
(t)]
2
+ [y
0
(t)]
2
dt.
ðÒÉÍÅÒ 7. ÷ÙÞÉÓÌÉÔØ ÄÌÉÎÕ ÁÓÔÒÏÉÄÙ x = 2 cos
3
t, y = 2 sin
3
t.
ôÁË ËÁË ËÒÉ×ÁÑ ÓÉÍÍÅÔÒÉÞÎÁ ÏÔÎÏÓÉÔÅÌØÎÏ ÏÂÅÉÈ ËÏÏÒÄÉÎÁÔÎÙÈ ÏÓÅÊ,
ÔÏ ×ÙÞÉÓÌÉÍ ÓÎÁÞÁÌÁ ÄÌÉÎÕ ÅÅ ÞÅÔ×ÅÒÔÏÊ ÞÁÓÔÉ l
1
, ÒÁÓÐÏÌÏÖÅÎÎÏÊ × ÐÅÒ×ÏÍ
Ë×ÁÄÒÁÎÔÅ, × ÜÔÏÍ ÓÌÕÞÁÅ 0 6 t 6
π
2
. x
0
t
= −6 cos
2
t · sin t, y
0
t
= 6 sin
2
t · cos t,
ÏÔÓÀÄÁ
l
1
=
π/2
Z
0
p
36 cos
4
t sin
2
t + 36 sin
4
t cos
2
t dt =
= 6
π/2
Z
0
p
cos
2
t sin
2
t dt = 6
π/2
Z
0
sin t cos t dt = 6
sin
2
t
2
π/2
0
=
6
2
= 3.
äÌÉÎÁ ×ÓÅÊ ËÒÉ×ÏÊ l = 4l
1
= 4 · 3 = 12.
ðÒÉÍÅÒ 8. ÷ÙÞÉÓÌÉÔØ ÄÌÉÎÕ ÃÉËÌÏÉÄÙ: x = (t − sin t), y = (1 − cos t),
t ∈ [0, 2π].
îÁÊÄÅÍ ÐÒÏÉÚ×ÏÄÎÙÅ x
0
t
= 1 − cos t, y
0
t
= sin t, ÔÏÇÄÁ
l =
2π
Z
0
q
(1 − cos t)
2
+ sin
2
t dt =
2π
Z
0
p
1 − 2 cos t + cos
2
t + sin
2
t dt =
=
2π
Z
0
√
2 − 2 cos t dt =
2π
Z
0
2 sin
t
2
dt = −4 cos
t
2
2π
0
=
= −4(cos π − cos 0) = −4(−1 − 1) = 8.
I.3. ëÒÉ×ÁÑ ÚÁÄÁÎÁ × ÐÏÌÑÒÎÙÈ ËÏÏÒÄÉÎÁÔÁÈ: r = r(ϕ), α 6 ϕ 6 β, ÔÏÇÄÁ
ÄÌÉÎÁ ËÒÉ×ÏÊ ×ÙÞÉÓÌÑÅÔÓÑ ÐÏ ÆÏÒÍÕÌÅ:
l =
β
Z
α
p
(r
0
(ϕ))
2
+ (r(ϕ))
2
dϕ.
ðÒÉÍÅÒ 9. ÷ÙÞÉÓÌÉÔØ ÄÌÉÎÕ ËÒÉ×ÏÊ r = (1 + cos ϕ), 0 6 ϕ 6 π.
§2. ïÐÒÅÄÅÌÅÎÎÙÊ ÉÎÔÅÇÒÁÌ, ÏÓÎÏ×ÎÙÅ Ó×ÏÊÓÔ×Á. . .                                                  41

  I.2. ëÒÉ×ÁÑ ÚÁÄÁÎÁ ÐÁÒÁÍÅÔÒÉÞÅÓËÉ: x = x(t), y = y(t), t ∈ [α, β], ÔÏÇÄÁ
ÄÌÉÎÁ ËÒÉ×ÏÊ ×ÙÞÉÓÌÑÅÔÓÑ ÐÏ ÆÏÒÍÕÌÅ:
                                   Zβ p
                                l=     [x0(t)]2 + [y 0 (t)]2 dt.
                                       α

   ðÒÉÍÅÒ 7. ÷ÙÞÉÓÌÉÔØ ÄÌÉÎÕ ÁÓÔÒÏÉÄÙ x = 2 cos3 t, y = 2 sin3 t.
   ôÁË ËÁË ËÒÉ×ÁÑ ÓÉÍÍÅÔÒÉÞÎÁ ÏÔÎÏÓÉÔÅÌØÎÏ ÏÂÅÉÈ ËÏÏÒÄÉÎÁÔÎÙÈ ÏÓÅÊ,
ÔÏ ×ÙÞÉÓÌÉÍ ÓÎÁÞÁÌÁ ÄÌÉÎÕ ÅÅ ÞÅÔ×ÅÒÔÏÊ ÞÁÓÔÉ l1, ÒÁÓÐÏÌÏÖÅÎÎÏÊ × ÐÅÒ×ÏÍ
Ë×ÁÄÒÁÎÔÅ, × ÜÔÏÍ ÓÌÕÞÁÅ 0 6 t 6 π2 . x0t = −6 cos2 t · sin t, yt0 = 6 sin2 t · cos t,
ÏÔÓÀÄÁ

      Zπ/2p
 l1 =      36 cos4 t sin2 t + 36 sin4 t cos2 t dt =
          0
                    Zπ/2p                       Zπ/2                                π/2
                                 2     2                            sin2 t                    6
               =6             cos t sin t dt = 6 sin t cos t dt = 6                       =     = 3.
                                                                      2             0         2
                       0                              0

äÌÉÎÁ ×ÓÅÊ ËÒÉ×ÏÊ l = 4l1 = 4 · 3 = 12.
   ðÒÉÍÅÒ 8. ÷ÙÞÉÓÌÉÔØ ÄÌÉÎÕ ÃÉËÌÏÉÄÙ: x = (t − sin t), y = (1 − cos t),
t ∈ [0, 2π].
   îÁÊÄÅÍ ÐÒÏÉÚ×ÏÄÎÙÅ x0t = 1 − cos t, yt0 = sin t, ÔÏÇÄÁ

    Z2π q                           Z2π p
 l=      (1 − cos t)2 + sin2 t dt =      1 − 2 cos t + cos2 t + sin2 t dt =
      0                                    0
                   Z2π                         Z2π                         2π
                           √                              t            t
               =            2 − 2 cos t dt =         2 sin dt = −4 cos          =
                                                          2            2   0
                   0                           0
                                                   = −4(cos π − cos 0) = −4(−1 − 1) = 8.
  I.3. ëÒÉ×ÁÑ ÚÁÄÁÎÁ × ÐÏÌÑÒÎÙÈ ËÏÏÒÄÉÎÁÔÁÈ: r = r(ϕ), α 6 ϕ 6 β, ÔÏÇÄÁ
ÄÌÉÎÁ ËÒÉ×ÏÊ ×ÙÞÉÓÌÑÅÔÓÑ ÐÏ ÆÏÒÍÕÌÅ:
                                  Zβ p
                               l=     (r0(ϕ))2 + (r(ϕ))2 dϕ.
                                   α

   ðÒÉÍÅÒ 9. ÷ÙÞÉÓÌÉÔØ ÄÌÉÎÕ ËÒÉ×ÏÊ r = (1 + cos ϕ), 0 6 ϕ 6 π.

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