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Консультация № 199924
20.12.2020, 12:50
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Помогите пожалуйста
Здравствуйте! Прошу помощи в следующем вопросе
Вычислить
┌┌
││( - 6x + 12y - 16)dxdy , где область D ограничена линиями
┘┘
D
x=7 ; y=7 ; x/7 + y/7 = 1:
Здравствуйте! Прошу помощи в следующем вопросе
Вычислить
┌┌
││( - 6x + 12y - 16)dxdy , где область D ограничена линиями
┘┘
D
x=7 ; y=7 ; x/7 + y/7 = 1:
Обсуждение
22.12.2020, 17:11
общий
это ответ
Здравствуйте, zhalykov.2015!
Условие : Функция f(x,y) = 12y - 6x - 16 ;
Область интегрирования ограничена прямыми : x = 7 ; y = 7 ; x + y = 7
Вычислить двойной интеграл I2 = D[$8747$][$8747$] f(x,y)·dx·dy
Решение : Строим график с 3мя заданными прямыми x = 7 ; y = 7 ; y = 7 - x , на котором достаточно наглядно видна область интегрирования. График прилагаю ниже. На графике я выделил область D голубой заливкой.
Приравниванием абсциссы прямых X1=7 и Y3(7) получаем ординату Y0 = 0 точки их пересечения (7 ; 0);
Приравниванием ординаты прямых Y2=7 и Y3(7 - x) получаем абсциссу X0 = 0 точки их пересечения (0 ; 7).
Все координаты области D получены.
Кто растерялся, с какого конца начинать решать двойные интегралы, читаем замечательную учебную статью "Двойные интегралы для чайников" Ссылка1 . Цитирую "Чтобы вычислить двойной интеграл, его надо свести к так называемым повторным интегралам"
Сначала интегрируем по "y" с порядком обхода от Y3(x) до Y2 . Вычисления и проверку я сделал в приложении Маткад (ссылка) . Маткад работает быстро и избавляет меня от частых ошибок. Маткад-скриншот прилагаю. Я добавил в него подробные комментарии зелёным цветом.
Ответ : Искомый двойной интеграл равен 294 .
Условие : Функция f(x,y) = 12y - 6x - 16 ;
Область интегрирования ограничена прямыми : x = 7 ; y = 7 ; x + y = 7
Вычислить двойной интеграл I2 = D[$8747$][$8747$] f(x,y)·dx·dy
Решение : Строим график с 3мя заданными прямыми x = 7 ; y = 7 ; y = 7 - x , на котором достаточно наглядно видна область интегрирования. График прилагаю ниже. На графике я выделил область D голубой заливкой.
Приравниванием абсциссы прямых X1=7 и Y3(7) получаем ординату Y0 = 0 точки их пересечения (7 ; 0);
Приравниванием ординаты прямых Y2=7 и Y3(7 - x) получаем абсциссу X0 = 0 точки их пересечения (0 ; 7).
Все координаты области D получены.
Кто растерялся, с какого конца начинать решать двойные интегралы, читаем замечательную учебную статью "Двойные интегралы для чайников" Ссылка1 . Цитирую "Чтобы вычислить двойной интеграл, его надо свести к так называемым повторным интегралам"
Сначала интегрируем по "y" с порядком обхода от Y3(x) до Y2 . Вычисления и проверку я сделал в приложении Маткад (ссылка) . Маткад работает быстро и избавляет меня от частых ошибок. Маткад-скриншот прилагаю. Я добавил в него подробные комментарии зелёным цветом.
Ответ : Искомый двойной интеграл равен 294 .
5
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