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Консультация № 177933
19.04.2010, 08:51
42.44 руб.
0
5
1
Добрый день. Помогите пожалуйста с примером:
Вычислить длину дуги данной линии:
x=[$8730$]3t2, y=t-t3, (-1[$8804$]t[$8804$]1)
Если можно подробней, заранее большое спасибо
Вычислить длину дуги данной линии:
x=[$8730$]3t2, y=t-t3, (-1[$8804$]t[$8804$]1)
Если можно подробней, заранее большое спасибо
Обсуждение
19.04.2010, 09:24
общий
L=-11[$8747$][$8730$]((x'(t))2+(y'(t))2)dt
Если я правильно понял
x=[$8730$]3*t2
y=t-t3
x'=[$8730$]3*2t
(x'(t))2=3*4*t2
y'=1-3t2
(y'(t))2=1-6t2+9t4
(x'(t))2+(y'(t))2=1-6t2+9t4+12*t2=9t4+6t2+1=(3t2+1)2
L=-11[$8747$][$8730$](3t2+1)2dt=-11[$8747$](3t2+1)dt=t3+t/-11=1+1+1+1=4
Если я правильно понял
x=[$8730$]3*t2
y=t-t3
x'=[$8730$]3*2t
(x'(t))2=3*4*t2
y'=1-3t2
(y'(t))2=1-6t2+9t4
(x'(t))2+(y'(t))2=1-6t2+9t4+12*t2=9t4+6t2+1=(3t2+1)2
L=-11[$8747$][$8730$](3t2+1)2dt=-11[$8747$](3t2+1)dt=t3+t/-11=1+1+1+1=4
19.04.2010, 09:48
общий
vera-nika:
Условие то я правильно прочитал или нет? Если что задавайте вопросы.
Условие то я правильно прочитал или нет? Если что задавайте вопросы.
Неизвестный
19.04.2010, 09:53
общий
vitalkise:
Условие вы правильно прочитали...вопросов у меня пока нет, жду когда лекции будут, буду вопросы преподавателю задавать
Условие вы правильно прочитали...вопросов у меня пока нет, жду когда лекции будут, буду вопросы преподавателю задавать
19.04.2010, 11:29
общий
это ответ
Здравствуйте, vera-nika.
По условию x(t) = √3t2, y(t) = t – t3. Находим производные по переменной t: dx/dt = 2√3t, dy/dt = 1 – 3t2. По формуле для нахождения длины кривой, заданной параметрически, получаем
-1∫1 √((2√3t)2 + (1 – 3t2)2)dt = -1∫1 √(12t2 + 1 – 6t2 + 9t4)dt = -1∫1 √(9t4 + 6t2 + 1)dt = -1∫1 √(3t2 + 1)2dt =
= -1∫1 (3t2 + 1)dt = (t3 + t)|-11 = (1 + 1) – (-1 + (-1)) = 4.
Ответ: 4.
С уважением.
По условию x(t) = √3t2, y(t) = t – t3. Находим производные по переменной t: dx/dt = 2√3t, dy/dt = 1 – 3t2. По формуле для нахождения длины кривой, заданной параметрически, получаем
-1∫1 √((2√3t)2 + (1 – 3t2)2)dt = -1∫1 √(12t2 + 1 – 6t2 + 9t4)dt = -1∫1 √(9t4 + 6t2 + 1)dt = -1∫1 √(3t2 + 1)2dt =
= -1∫1 (3t2 + 1)dt = (t3 + t)|-11 = (1 + 1) – (-1 + (-1)) = 4.
Ответ: 4.
С уважением.
5
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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