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Консультация № 201470
10.10.2021, 14:26
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1
Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:
Вычислить площадь фигуры, ограниченной графиками функций.
Дано: y = cosx(sin^2)x; x,y=0; 0<=x<=П/2
Вычислить площадь фигуры, ограниченной графиками функций.
Дано: y = cosx(sin^2)x; x,y=0; 0<=x<=П/2
Обсуждение
11.10.2021, 02:23
общий
Адресаты:
Хорошо, что Вы написали Условие текстом. Но Ваше выражение "cosx(sin^2)x;" - это полная бессмыслица, означающая Ваше НЕжелание вникнуть в смысл Условия. Ваше неуважение к задаче и изучаемому предмету побуждает не помогать Вам в Решении, а наоборот: посоветовать Вам забросить ненавистное занятие и переключиться на то, что Вы умеете и любите делать.
Я окультурил Ваши мат-выражения и построил графики. Вам осталось вычислить площадь голубой заливки.
Дальше сами сможете?
Я окультурил Ваши мат-выражения и построил графики. Вам осталось вычислить площадь голубой заливки.
Дальше сами сможете?
Прикрепленные файлы:
12.10.2021, 04:45
общий
это ответ
Условие : Даны 2 функции y1(x) = cos(x)·sin2(x) , y2(x) = 0 и отрезок x[0 ; [$960$]/2] .
Вычислить площадь фигуры, ограниченной графиками функций на заданном отрезке.
Решение : Читаем учебную статью "Определённый интеграл. Как вычислить площадь фигуры ?" Ссылка1 и вычисляем искомую площадь по простейшей формуле
S = x1x2[$8747$] y1(x)·dx = 0[$960$]/2[$8747$] cos(x)·sin2(x)·dx = 0[$960$]/2[$8747$] sin2(x)·d[sin(x)] = [sin3(x) / 3] |0[$960$]/2 =
= (1/3)·[sin3([$960$]/2) - sin3(0)] = (1/3)·(13 - 03) = (1/3)·1 = 1/3 [$8776$] 0,33
В процессе интегрирования использован приём замены cos(x)·dx на d[sin(x)], называемый "Подведение функции под знак дифференциала" на том основани, что
[sin(x)]' = d[sin(x)] / dx = cos(x) (см "Метод замены переменной в неопределенном интеграле. Примеры решений" Ссылка2
Ответ: площадь фигуры равна 1/3 [$8776$] 0,33 ед2.
Для проверки Вы можете сверить вычисленную площадь с суммарной площадью голубой заливки на графике (приложен выше) с помощью координатной сетки. Площадь одного прямоугольничка сетки равна 0,5·0,1 = 0,05 ед2. =Удачи!
Вычислить площадь фигуры, ограниченной графиками функций на заданном отрезке.
Решение : Читаем учебную статью "Определённый интеграл. Как вычислить площадь фигуры ?" Ссылка1 и вычисляем искомую площадь по простейшей формуле
S = x1x2[$8747$] y1(x)·dx = 0[$960$]/2[$8747$] cos(x)·sin2(x)·dx = 0[$960$]/2[$8747$] sin2(x)·d[sin(x)] = [sin3(x) / 3] |0[$960$]/2 =
= (1/3)·[sin3([$960$]/2) - sin3(0)] = (1/3)·(13 - 03) = (1/3)·1 = 1/3 [$8776$] 0,33
В процессе интегрирования использован приём замены cos(x)·dx на d[sin(x)], называемый "Подведение функции под знак дифференциала" на том основани, что
[sin(x)]' = d[sin(x)] / dx = cos(x) (см "Метод замены переменной в неопределенном интеграле. Примеры решений" Ссылка2
Ответ: площадь фигуры равна 1/3 [$8776$] 0,33 ед2.
Для проверки Вы можете сверить вычисленную площадь с суммарной площадью голубой заливки на графике (приложен выше) с помощью координатной сетки. Площадь одного прямоугольничка сетки равна 0,5·0,1 = 0,05 ед2. =Удачи!
5
Спасибо!!!
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