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Консультация № 180041
26.09.2010, 12:31
0.00 руб.
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Здравствуйте эксперты! Помогите решить задачу: Вычислить криволинейный интеграл 1-го рода:
∫y^2 dl, в области С, где С – арка циклоид ;
x=a(t-sint); y=a(1-cost); (0<=t<=2π);
Заранее Большое Спасибо
∫y^2 dl, в области С, где С – арка циклоид ;
x=a(t-sint); y=a(1-cost); (0<=t<=2π);
Заранее Большое Спасибо
Обсуждение
Неизвестный
26.09.2010, 17:23
общий
Здравствуйте, Magma! Вычислим данный криволинейный интеграл 1 рода, рассматривая только 1 арку циклоиды:
dl=[$8730$]((x')2+(y')2)dt = [$8730$]((a[$215$](1-cos(t)))2+(a[$215$]sin(t))2)dt=a[$215$][$8730$](2-2[$215$]cos(t))dt=a[$8730$]2[$215$](1-cos(t))dt
тогда, получим следующий интеграл: [$8747$] (от 0 до 2pi)a2[$215$](1-cos(t))2[$215$]a[$8730$]2[$215$](1-cos(t))dt=a3[$8730$]2[$8747$] (от 0 до 2pi)(1-cos(t))5/2dt=a3[$8730$]2[$8747$] (от 0 до 2pi)(1- (1-tg(t/2)2)/(1+tg(t/2)2))5/2dt=a3[$8730$]2[$8747$] (от 0 до 2pi)(2[$215$]tg(t/2)2/(1+tg(t/2)2))5/2dt=a3[$8730$]2[$8747$] (от 0 до 2pi)(2[$215$]sin(t/2))5dt=8a3[$8747$] (от 0 до 2pi)(sin(t/2))5dt=8a3[$8747$] (от 0 до 2pi)(sin(t/2))4[$215$](sin(t/2)dt=-16a3[$8747$] (от 0 до 2pi)(1-(cos(t/2))2)2d(cos(t/2))=16a3[$215$](-cos(t/2)+2/3[$215$](cos(t/2))3-1/5[$215$](cos(t/2))5)|(от 0 до 2pi)=16a3[$215$]16/15=256/15a3.
dl=[$8730$]((x')2+(y')2)dt = [$8730$]((a[$215$](1-cos(t)))2+(a[$215$]sin(t))2)dt=a[$215$][$8730$](2-2[$215$]cos(t))dt=a[$8730$]2[$215$](1-cos(t))dt
тогда, получим следующий интеграл: [$8747$] (от 0 до 2pi)a2[$215$](1-cos(t))2[$215$]a[$8730$]2[$215$](1-cos(t))dt=a3[$8730$]2[$8747$] (от 0 до 2pi)(1-cos(t))5/2dt=a3[$8730$]2[$8747$] (от 0 до 2pi)(1- (1-tg(t/2)2)/(1+tg(t/2)2))5/2dt=a3[$8730$]2[$8747$] (от 0 до 2pi)(2[$215$]tg(t/2)2/(1+tg(t/2)2))5/2dt=a3[$8730$]2[$8747$] (от 0 до 2pi)(2[$215$]sin(t/2))5dt=8a3[$8747$] (от 0 до 2pi)(sin(t/2))5dt=8a3[$8747$] (от 0 до 2pi)(sin(t/2))4[$215$](sin(t/2)dt=-16a3[$8747$] (от 0 до 2pi)(1-(cos(t/2))2)2d(cos(t/2))=16a3[$215$](-cos(t/2)+2/3[$215$](cos(t/2))3-1/5[$215$](cos(t/2))5)|(от 0 до 2pi)=16a3[$215$]16/15=256/15a3.
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