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Консультация № 142116
24.08.2008, 17:33
0.00 руб.
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Здравствуйте. Помогите решить 2 задачи:
1) Вычислить площадь фигуры, ограниченной линиями: y=lg(1/x), y=x-1, x=10.
lg - логарифм по основанию 10
2) Вычислить объем тела, ограниченного поверхностями: x - y /4+ z /4=1, y=0, y=2.
Когда будете интегрировать, распишите подробнее.
1) Вычислить площадь фигуры, ограниченной линиями: y=lg(1/x), y=x-1, x=10.
lg - логарифм по основанию 10
2) Вычислить объем тела, ограниченного поверхностями: x - y /4+ z /4=1, y=0, y=2.
Когда будете интегрировать, распишите подробнее.
Обсуждение
27.08.2008, 08:52
общий
Могу помочь наметить линию решения.
Прежде всего, нужен примерный график на декартовой плоскости, где прорисованы данные линии.
Из чертежа будет видно, что фигура заключена между кривыми у=х-1 (сверху) и lg(1/х) снизу.
Слева и справа она ограничена прямыми х=1 и х=10.
Отсюда искомый интеграл:
S=Int[1;10] (x-1-lg(1/x)) dx
Сложность представляет последнее выражение под интегралом, но оно сначала преобразуется к lg(x), ln(x), а затем заменой
x=exp(t);
dx=exp(t) dt
x э [1;10] -> t э [0; ln(10)]
приводится к интегралу, который последовательным интегрированием по частям также легко решается.
Прежде всего, нужен примерный график на декартовой плоскости, где прорисованы данные линии.
Из чертежа будет видно, что фигура заключена между кривыми у=х-1 (сверху) и lg(1/х) снизу.
Слева и справа она ограничена прямыми х=1 и х=10.
Отсюда искомый интеграл:
S=Int[1;10] (x-1-lg(1/x)) dx
Сложность представляет последнее выражение под интегралом, но оно сначала преобразуется к lg(x), ln(x), а затем заменой
x=exp(t);
dx=exp(t) dt
x э [1;10] -> t э [0; ln(10)]
приводится к интегралу, который последовательным интегрированием по частям также легко решается.
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