Презентация к уроку "Интеграл"

Интеграл

ИНТЕГРАЛ

Неопределённый

интеграл

Определённый

интеграл

Обозначение:

Определение:

Множество всех первообразных функции f(x) на некотором промежутке называется неопределенным интегралом от функции f(x) на этом промежутке и обозначается

Основные свойства неопределенного интеграла.

Таблица интегралов

Определение

• Процесс нахождения интеграла называется  интегрированием.
• Интегрирование является операцией, обратной  дифференцированию

ОПРЕДЕЛЁННЫЙ

ИНТЕГРАЛ

Определение

Пусть функция y=f(x) определена и интегрируема на отрезке [ a,b ] и пусть F(x) – некоторая ее первообразная. Тогда число F(b)–F(a) называется интегралом от а до b функции f(x) и обозначается

Определение

Криволинейной трапецией называется фигура, ограниченная

графиком функции f(x) ,

графиками х=а и х=в , и осью ОХ

а

в

Формула Ньютона-Лейбница

Теорема: если функция у = f(x) непрерывна на отрезке [ а;в ] , то справедлива формула

Опираясь на эту формулу получаются следующие свойства определенного интеграла

Свойства определенного интеграла

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Свойства определенного интеграла

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Геометрический смысл определенного интеграла – это площадь криволинейной трапеции

Вычисление объема тела вращения

Объем тела , образованного вращением вокруг оси Ox криволинейной трапеции , ограниченной кривой

у = f(x) отрезком оси абсцисс a ≤ x ≤ b и прямыми x=a; x=b , вычисляется по формуле