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Консультация № 180267
10.10.2010, 16:22
50.00 руб.
0
2
2
Уважаемые эксперты, помогите решить пример 
:
Вычислить площадь фигуры, ограниченной линиями:
x=2(t-sin (t))
y=2(1-cos(t))
y=3
при y[$8805$]3 и 0[$8804$]x[$8804$]4п
:Вычислить площадь фигуры, ограниченной линиями:
x=2(t-sin (t))
y=2(1-cos(t))
y=3
при y[$8805$]3 и 0[$8804$]x[$8804$]4п
Обсуждение
Неизвестный
10.10.2010, 16:56
общий
это ответ
Здравствуйте, Николай Алексеевич.
Имеем циклоиду с а=2 . У изменяется от 0 до 2а , то есть от 0 до 6 . Х изменяется от 0 до 2*а*Pi , то есть в нашем распоряжении 1 арка этой циклоиды . S=?(y(t))*(((x(t))')*dt , 0<t<4*Pi .
x=2(t-sin t)=>x'=2(1-cost)dt
S=?2(1-cost)*2(1-cost)dt=4*?(1-2*cost+((cost)^2))*dt=4*?*dt-8*?costdt+
+4*?(1/2)*(1+cos2t)*dt=4*t-8*sint+2*t+sin2t=...
{ Теперь осталось подставить пределы интегрирования . Помним что у синуса период 2пи и сразу считаем все выражения с синусом как 0 }
...=6*t=6*(2*Pi-0)=12*Pi .
OTBET : S=12*Pi квадратных едениц площади ( приблизительно 37.7 ) .
P.S. Простите что без рисунка , но зная название "циклоида" её рисунок можно найти в любой мало-мальски приличной литературе .
Удачи .
Имеем циклоиду с а=2 . У изменяется от 0 до 2а , то есть от 0 до 6 . Х изменяется от 0 до 2*а*Pi , то есть в нашем распоряжении 1 арка этой циклоиды . S=?(y(t))*(((x(t))')*dt , 0<t<4*Pi .
x=2(t-sin t)=>x'=2(1-cost)dt
S=?2(1-cost)*2(1-cost)dt=4*?(1-2*cost+((cost)^2))*dt=4*?*dt-8*?costdt+
+4*?(1/2)*(1+cos2t)*dt=4*t-8*sint+2*t+sin2t=...
{ Теперь осталось подставить пределы интегрирования . Помним что у синуса период 2пи и сразу считаем все выражения с синусом как 0 }
...=6*t=6*(2*Pi-0)=12*Pi .
OTBET : S=12*Pi квадратных едениц площади ( приблизительно 37.7 ) .
P.S. Простите что без рисунка , но зная название "циклоида" её рисунок можно найти в любой мало-мальски приличной литературе .
Удачи .
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