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Консультация № 68967
28.12.2006, 21:12
0.00 руб.
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Уважаемые эксперты. Помогите составить интеграл. Чертеж начертил, а интеграл не могу составить.
1) Вычислить объем тел x^2/3+y^2/4=1; z=y3^1/2; z=6 (z>=6)
2) Вычислить объем тел вращения фигур y=x^2 и y=(x+1)^1/2
Нужен только интеграл, дальше сам справлюсь.
Заранее благодарен.
1) Вычислить объем тел x^2/3+y^2/4=1; z=y3^1/2; z=6 (z>=6)
2) Вычислить объем тел вращения фигур y=x^2 и y=(x+1)^1/2
Нужен только интеграл, дальше сам справлюсь.
Заранее благодарен.
Обсуждение
Неизвестный
29.12.2006, 10:59
общий
это ответ
Здравствуйте, Калимуллин Дамир Рустамович!
Полученное тело разбиваем на 2 плоскостью z = 2*3^0.5 (самая верхняя точка пересечения z = y*3^0.5 и x^2/3 + y^2/4 = 1, она получается при y = 2).
Объем верхней части равен площади полуэллипса, умноженной на высоту h = 6 - 2*3^0.5
Для нижней составляем интеграл:
инт(от 0 до 2*3^0.5)dz инт(от 0 до z/3^0.5)dy инт(от -sqrt(3 - 3*y^2/4) до sqrt(3 - 3*y^2/4)) dx.
Также предположу, что можно получить более легкий интеграл через цилиндрические координаты, но я не помню Якобиана.
Насчет второго, могу лишь предположить, что здесь нужно использовать цилиндрические координаты. Точного решения увы не знаю.
Полученное тело разбиваем на 2 плоскостью z = 2*3^0.5 (самая верхняя точка пересечения z = y*3^0.5 и x^2/3 + y^2/4 = 1, она получается при y = 2).
Объем верхней части равен площади полуэллипса, умноженной на высоту h = 6 - 2*3^0.5
Для нижней составляем интеграл:
инт(от 0 до 2*3^0.5)dz инт(от 0 до z/3^0.5)dy инт(от -sqrt(3 - 3*y^2/4) до sqrt(3 - 3*y^2/4)) dx.
Также предположу, что можно получить более легкий интеграл через цилиндрические координаты, но я не помню Якобиана.
Насчет второго, могу лишь предположить, что здесь нужно использовать цилиндрические координаты. Точного решения увы не знаю.
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