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Консультация № 109055
12.11.2007, 13:22
0.00 руб.
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Здравствуйте, уважаемые эксперты. Помогите решить задачу.
Вычислить с помощью тройного интеграла объем тела, огранниченного поверхностями z=0,z=1-y^2, x^2=y^2, x=2y^2+1 и сделать чертеж. Студентка Наташа Бондарчук
Вычислить с помощью тройного интеграла объем тела, огранниченного поверхностями z=0,z=1-y^2, x^2=y^2, x=2y^2+1 и сделать чертеж. Студентка Наташа Бондарчук
Обсуждение
Неизвестный
17.11.2007, 11:14
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это ответ
Здравствуйте, Бондарчук Наташа Владимировна!
Здравствуйте, уважаемые эксперты. Помогите решить задачу.
Вычислить с помощью тройного интеграла объем тела, огранниченного поверхностями z=0,z=1-y^2, x^2=y^2, x=2y^2+1 и сделать чертеж. Студентка Наташа Бондарчук
Поверхности z = 0 и z = 1 - y<sup>2</sup> определяют чать параболического цилиндра с осью вдоль оси х. Граница пересечения y = +-1.
x<sup>2</sup> = y<sup>2</sup> - определяет две плоскости: y = x и y = -x.
x = 1 + 2y<sup>2</sup> - параболический цилиндр с осью вдоль оси z.
Удобно завершать интегрировать по переменной y, которая может меняться от -1 до 1, поскольку должна находиться внутри цилиндра радиуса 1.
Тогда z меняется от 0 до 1-y<sup>2</sup> и x меняется от |y| до 1 + 2y<sup>2</sup>.
Из симметрии относительно оси x мы можем посчитать удвоенный интеграл для y в пределе от 0 до 1, тогда пределы для переменной x будут от y до 1 + 2y<sup>2</sup>.
Окончательно,
V = 2*∫(0,1,∫(y,1 + 2y<sup>2</sup>,∫(0, 1-y<sup>2</sup>,dz)dx)dy =
= 2*∫(0,1,∫(y,1 + 2y<sup>2</sup>,(1-y<sup>2</sup>)dx)dy =
= 2*∫(0,1,(1-y<sup>2</sup>)(1 - y + 2y<sup>2</sup>)dy =
= 2*∫(0,1, 1 - y + y<sup>2</sup> + y<sup>3</sup> - 2y<sup>4</sup>)dy =
= 2*(1 - 1/2 + 1/3 + 1/4 - 2*1/5)= 41/30
Проверьте вычисления
Здравствуйте, уважаемые эксперты. Помогите решить задачу.
Вычислить с помощью тройного интеграла объем тела, огранниченного поверхностями z=0,z=1-y^2, x^2=y^2, x=2y^2+1 и сделать чертеж. Студентка Наташа Бондарчук
Поверхности z = 0 и z = 1 - y<sup>2</sup> определяют чать параболического цилиндра с осью вдоль оси х. Граница пересечения y = +-1.
x<sup>2</sup> = y<sup>2</sup> - определяет две плоскости: y = x и y = -x.
x = 1 + 2y<sup>2</sup> - параболический цилиндр с осью вдоль оси z.
Удобно завершать интегрировать по переменной y, которая может меняться от -1 до 1, поскольку должна находиться внутри цилиндра радиуса 1.
Тогда z меняется от 0 до 1-y<sup>2</sup> и x меняется от |y| до 1 + 2y<sup>2</sup>.
Из симметрии относительно оси x мы можем посчитать удвоенный интеграл для y в пределе от 0 до 1, тогда пределы для переменной x будут от y до 1 + 2y<sup>2</sup>.
Окончательно,
V = 2*∫(0,1,∫(y,1 + 2y<sup>2</sup>,∫(0, 1-y<sup>2</sup>,dz)dx)dy =
= 2*∫(0,1,∫(y,1 + 2y<sup>2</sup>,(1-y<sup>2</sup>)dx)dy =
= 2*∫(0,1,(1-y<sup>2</sup>)(1 - y + 2y<sup>2</sup>)dy =
= 2*∫(0,1, 1 - y + y<sup>2</sup> + y<sup>3</sup> - 2y<sup>4</sup>)dy =
= 2*(1 - 1/2 + 1/3 + 1/4 - 2*1/5)= 41/30
Проверьте вычисления
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