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Консультация № 189562
08.06.2016, 22:01
0.00 руб.
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Здравствуйте! Прошу помощи в следующем вопросе:
Задание. 1) Вычислить интеграл по формуле трапеций с тремя десятичными знаками после запятой.
2) Вычислить интеграл по формуле Симпсона при n=8 . оценить погрешность результата, составив таблицу конечных разностей.
сам пример
Задание. 1) Вычислить интеграл по формуле трапеций с тремя десятичными знаками после запятой.
2) Вычислить интеграл по формуле Симпсона при n=8 . оценить погрешность результата, составив таблицу конечных разностей.
сам пример
Обсуждение
10.06.2016, 11:11
общий
это ответ
Здравствуйте, poijkl2016!
Вычислим интеграл аналитически:
Вычислим интеграл по формуле трапеций с тремя десятичными знаками после запятой. Для простоты полагаем, что точность
вычисления интеграла по формуле трапеций связана с шагом 
интегрирования зависимостью 
Если принять ориентировочно 
то 
Примем 
и 
Тогда отрезок интегрирования будет разбит на 
частей. Выполним необходимые вычисления, используя приложение MS Excel. В результате по формуле трапеций получим
Вычислим интеграл по формуле Симпсона при
Выполним необходимые вычисления, используя приложение MS Excel. В результате по формуле Симпсона получим
Для оценки погрешности полученного результата составим в том же приложении таблицу конечных разностей до четвёртого порядка включительно. Из таблицы видно, что максимальное значение модуля конечной разности четвёртого порядка составляет
Следовательно, остаточный член равен
Правда, это не соответствует действительности, в чём можно убедиться, сравнив значение интеграла, полученное аналитически, со значением, полученным по формуле Симпсона.
Вычислим интеграл аналитически:
Вычислим интеграл по формуле трапеций с тремя десятичными знаками после запятой. Для простоты полагаем, что точность
вычисления интеграла по формуле трапеций связана с шагом интегрирования зависимостью Если принять ориентировочно то Примем и Тогда отрезок интегрирования будет разбит на частей. Выполним необходимые вычисления, используя приложение MS Excel. В результате по формуле трапеций получимВычислим интеграл по формуле Симпсона при
Выполним необходимые вычисления, используя приложение MS Excel. В результате по формуле Симпсона получимДля оценки погрешности полученного результата составим в том же приложении таблицу конечных разностей до четвёртого порядка включительно. Из таблицы видно, что максимальное значение модуля конечной разности четвёртого порядка составляет
Следовательно, остаточный член равенПравда, это не соответствует действительности, в чём можно убедиться, сравнив значение интеграла, полученное аналитически, со значением, полученным по формуле Симпсона.
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Об авторе:
Facta loquuntur.
Facta loquuntur.
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