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Консультация № 164609
10.04.2009, 17:57
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Доброго времени суток, математики! Прошу помочь в решении следующей задачи:
С помощью тройного интеграла найти объем тела, ограниченного указанными поверхностями
y=6*sqrt(3*x), y=sqrt(3*x), z=0, x+z=3
Спасибо
С помощью тройного интеграла найти объем тела, ограниченного указанными поверхностями
y=6*sqrt(3*x), y=sqrt(3*x), z=0, x+z=3
Спасибо
Обсуждение
Неизвестный
10.04.2009, 21:54
общий
это ответ
Здравствуйте, Evgenya!
Построив чертеж, найдем пределы изменения каждого значения.
Во-первых, из областей определения и множества значений функций y=6*sqrt(3*x), y=sqrt(3*x) следует, что x>=0 и y>=0. Потом из уравнения x+z=3 и условия, x>=0 следует, что z<=3. В то же время, по условию, z>=0.
Итак, пределы изменения z найдены: [0,3].
Фиксируем некоторое значение z = z0. Тогда из уравнения x+z=3 следует, что x меняется от 0 до 3-z0 (нижняя граница (0) была найдена ранее). Т.к. z0 пробегает все возможные значения z, то можно смело утверждять, что пределы изменения x есть [0, 3-z].
Далее из равенств y=6*sqrt(3*x), y=sqrt(3*x) следует, что пределы измеения y есть [6*sqrt(3*x), sqrt(3*x)].
Теперь можно вычислить объем через тройные интегралы
V = ∫[0,3]dz∫[0,3-z]dx∫[sqrt(3*x),6*sqrt(3*x)]dy = ∫[0,3]dz∫[0,3-z](5*sqrt(3*x))dx = (10*sqrt(3)/3) * ∫[0,3]((z-3)^(3/2))dz = 36.
Ответ: 36.
Построив чертеж, найдем пределы изменения каждого значения.
Во-первых, из областей определения и множества значений функций y=6*sqrt(3*x), y=sqrt(3*x) следует, что x>=0 и y>=0. Потом из уравнения x+z=3 и условия, x>=0 следует, что z<=3. В то же время, по условию, z>=0.
Итак, пределы изменения z найдены: [0,3].
Фиксируем некоторое значение z = z0. Тогда из уравнения x+z=3 следует, что x меняется от 0 до 3-z0 (нижняя граница (0) была найдена ранее). Т.к. z0 пробегает все возможные значения z, то можно смело утверждять, что пределы изменения x есть [0, 3-z].
Далее из равенств y=6*sqrt(3*x), y=sqrt(3*x) следует, что пределы измеения y есть [6*sqrt(3*x), sqrt(3*x)].
Теперь можно вычислить объем через тройные интегралы
V = ∫[0,3]dz∫[0,3-z]dx∫[sqrt(3*x),6*sqrt(3*x)]dy = ∫[0,3]dz∫[0,3-z](5*sqrt(3*x))dx = (10*sqrt(3)/3) * ∫[0,3]((z-3)^(3/2))dz = 36.
Ответ: 36.
Форма ответа
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