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Консультация № 198200

10.04.2020, 09:35

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Образование

Здравствуйте! Прошу помощи в следующем вопросе:
вычислить площадь фигуры, ограниченной графиком функции
y=x/(x^2+1)^2, y=0, x=1

Заранее, спасибо)

Обсуждение

давно

Алексеев Владимир Николаевич

Мастер-Эксперт
259041
7459

13.04.2020, 15:47

общий

это ответ

Здравствуйте, goroshinka!
Дано : y1(x) = x / (x² + 1)² , y2 = 0 , x3=1
Вычислить площадь м-ду тремя функциями.

Решение : Сначала строим графики всех 3х функций. Рисунок прилагаю ниже.
На графике очевидно, что искомая площадь представляет собой область, залитую голубым цветом.
Вычислить эту область удобнее всего интегрированием функции y1(x) = x / (x² + 1)² в интервале от x0=0 до x3=1.
Точка x0 найдена пересечением функций y1(x) и y2 = 0 , то есть решением уравнения y1(x0) = 0.
Ответ : искомая площадь равна 0,25 кв.единиц.

Упрощённая проверка : Площадь одного квадратика зелёной сетки на графике равна 1·0,1 = 0,1 кв.единиц.
Залитая область занимает примерно 2,5 квадратика. Значит, искомая площадь равна 2,5·0,1 = 0,25 кв.единиц. Проверка успешна.

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