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Консультация № 195299
18.04.2019, 04:03
0.00 руб.
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Здравствуйте! У меня возникли сложности с таким вопросом:
Найти объем тела, ограниченного поверхностями
(параболоид) и 
(конус).
Найти объем тела, ограниченного поверхностями
(параболоид) и (конус).
Обсуждение
22.04.2019, 08:57
общий
это ответ
Здравствуйте, bolanebyla!
Перепишем уравнения поверхностей в виде:
x2/2z +y2/8z = 1 (параболоид), x2/z2 +y2/4z2 = 1 (конус).
В сечении плоскостью z = const - это уравнения эллипса, с полуосями
a=2[$8730$](2z), b = [$8730$](2z) для параболоида и a=2z, b = z для конуса.
При z=2 эллипсы совпадают, при 0 [$8804$] z [$8804$] 2 второй лежит внутри первого.
Вспоминая, что площадь эллипса равна pi ab, найдем площадь фигуры,
заключенной между эллипсами:
S(z) = pi (4z - 2z2).
Искомый объем найдется как интеграл по z в пределах от 0 до 2:
V = [$8747$]S(z)dz = 2pi [$8747$](2z - z2)dz = 2pi (z2-z3/3)|20 = 8pi/3.
Перепишем уравнения поверхностей в виде:
x2/2z +y2/8z = 1 (параболоид), x2/z2 +y2/4z2 = 1 (конус).
В сечении плоскостью z = const - это уравнения эллипса, с полуосями
a=2[$8730$](2z), b = [$8730$](2z) для параболоида и a=2z, b = z для конуса.
При z=2 эллипсы совпадают, при 0 [$8804$] z [$8804$] 2 второй лежит внутри первого.
Вспоминая, что площадь эллипса равна pi ab, найдем площадь фигуры,
заключенной между эллипсами:
S(z) = pi (4z - 2z2).
Искомый объем найдется как интеграл по z в пределах от 0 до 2:
V = [$8747$]S(z)dz = 2pi [$8747$](2z - z2)dz = 2pi (z2-z3/3)|20 = 8pi/3.
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