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Задать вопрос экспертам

Консультация № 187877

07.06.2014, 15:47

90.00 руб.

0 1 1

Образование

Здравствуйте, уважаемые эксперты! Прошу вас ответить на следующий вопрос:

http://dfiles.ru/files/bxev77wwo
Задача 8.

Обсуждение

давно

Roman Chaplinsky / Химик CH

Модератор
156417
2175

07.06.2014, 17:01

общий

это ответ

Здравствуйте, Aleksandrkib!
Вычислить объём тела, образованного вращением вокруг оси Ох фигуры, ограниченной прямой x-y+1=0 дугой косинусоиды y = cos x и осью Ох

Образующая тела вращения задана уравнением
[TNR]y(x)=x+1 при x[$8712$][-1;0]
y(x)=cos(x) при x[$8712$][0;[$960$]/2][/TNR]

каждый точечный отрезок длины [TNR]dx[/TNR] возле точки с координатой [TNR]x[/TNR] ограничивает цилиндрическую область пространства радиусом [TNR]y(x)[/TNR] и объёмом [TNR]dV=[$960$]y(x)²dx[/TNR]
объём находится по формуле [TNR]V=_{x min}^{x max}[$8747$][$960$]y(x)²dx[/TNR]
Для исследуемой функции этот интеграл принимает форму
[TNR]V=[$960$](_-1⁰[$8747$](x+1)²dx+₀^[$960$]/2[$8747$]cos²(x)dx)=
=[$960$](1/3[$183$](x+1)³|_-1⁰+(x/2+1/4[$183$]sin(2x))|₀^[$960$]/2)=
=[$960$](1/3[$183$]((0+1)³-(-1+1)³)+([$960$]/4+1/4[$183$]sin([$960$])-0/2-1/4[$183$]sin(0)))=
=[$960$](1/3+[$960$]/4)=[$960$]/3+[$960$]²/4[/TNR]

5

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