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Консультация № 192030
07.12.2017, 11:54
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Уважаемые эксперты! Пожалуйста, ответьте на вопрос:
Вычислить объем тела вращения вокруг оси OY фигуры, ограниченной графиками функций y=x^2, y=0, x=2.
Вычислить объем тела вращения вокруг оси OY фигуры, ограниченной графиками функций y=x^2, y=0, x=2.
Обсуждение
17.12.2017, 04:57
общий
это ответ
Здравствуйте, sigma5696!
В общем случае объем тела вращения определяется по одной из двух формул:
- если фигура вращается вокруг оси Ox и ограничена кривыми y[sub]1[/sub](x), y[sub]2[/sub](x) и прямыми x = x[sub]1[/sub], x = x[sub]2[/sub] (причём y[sub]1[/sub] < y[sub]2[/sub] при x[sub]1[/sub] < x < x[sub]2[/sub]);
- если фигура вращается вокруг оси Oy и ограничена кривыми x[sub]1[/sub](y), x[sub]2[/sub](y) и прямыми y = y[sub]1[/sub], y = y[sub]2[/sub] (причём x[sub]1[/sub] < x[sub]2[/sub] при y[sub]1[/sub] < y < y[sub]2[/sub]).
В данном случае имеем x[sub]1[/sub](y) = [$8730$]y (функция, обратная y = x[sup]2[/sup]), x[sub]2[/sub] = 2, y[sub]1[/sub] = 0, y[sub]2[/sub] = 4 (точка пересечения кривой y=x[sup]2[/sup] и прямой x=2), поэтому объем тела вращения будет равен
В общем случае объем тела вращения определяется по одной из двух формул:
- если фигура вращается вокруг оси Ox и ограничена кривыми y[sub]1[/sub](x), y[sub]2[/sub](x) и прямыми x = x[sub]1[/sub], x = x[sub]2[/sub] (причём y[sub]1[/sub] < y[sub]2[/sub] при x[sub]1[/sub] < x < x[sub]2[/sub]);
- если фигура вращается вокруг оси Oy и ограничена кривыми x[sub]1[/sub](y), x[sub]2[/sub](y) и прямыми y = y[sub]1[/sub], y = y[sub]2[/sub] (причём x[sub]1[/sub] < x[sub]2[/sub] при y[sub]1[/sub] < y < y[sub]2[/sub]).
В данном случае имеем x[sub]1[/sub](y) = [$8730$]y (функция, обратная y = x[sup]2[/sup]), x[sub]2[/sub] = 2, y[sub]1[/sub] = 0, y[sub]2[/sub] = 4 (точка пересечения кривой y=x[sup]2[/sup] и прямой x=2), поэтому объем тела вращения будет равен
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