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Консультация № 55073
11.09.2006, 20:04
0.00 руб.
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1
Найти площадь фигуры, ограниченую графиками функций
x=6*cos(t)
y=4*sin(t)
x=6*cos(t)
y=4*sin(t)
Обсуждение
12.09.2006, 12:31
общий
это ответ
Здравствуйте, Mr Jackal!
Используем симметрию заданной фигуры. Найдём площадь части фигуры, расположенной в первом квадранте, и учетверим её. Точка (0; 4) получается при t= π/2, точка (6; 0) - при t = 0; x׳(t)=6∙sin (t), поэтому
S=4∙∫(0, π/2) 4∙sin (t)∙(-6∙sin (t))dt=-96∙∫(0, π/2) (sin (t))^2 dt.
Воспользуемся формулой
Int (sin x)^2 dx = x/2-(sin 2x)/4,
тогда
∫(0, π/2) (sin (t))^2 dt= x/2-(sin 2x)/4|(0, π/2)=-π/4,
S=-96∙(-π/4)=24∙π.
Полученный ответ совпадает с получаемым по известной из аналитической геометрии формуле S= π∙a∙b= π∙6∙4=24∙π.
Используем симметрию заданной фигуры. Найдём площадь части фигуры, расположенной в первом квадранте, и учетверим её. Точка (0; 4) получается при t= π/2, точка (6; 0) - при t = 0; x׳(t)=6∙sin (t), поэтому
S=4∙∫(0, π/2) 4∙sin (t)∙(-6∙sin (t))dt=-96∙∫(0, π/2) (sin (t))^2 dt.
Воспользуемся формулой
Int (sin x)^2 dx = x/2-(sin 2x)/4,
тогда
∫(0, π/2) (sin (t))^2 dt= x/2-(sin 2x)/4|(0, π/2)=-π/4,
S=-96∙(-π/4)=24∙π.
Полученный ответ совпадает с получаемым по известной из аналитической геометрии формуле S= π∙a∙b= π∙6∙4=24∙π.
Об авторе:
Facta loquuntur.
Facta loquuntur.
Форма ответа
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