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Консультация № 176491
04.02.2010, 16:01
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Здравствуйте уважаемые эксперты помогите решить еще одну задачку
нужно Определить площадь плоской фигуры, ограниченной кривыми
нужно Определить площадь плоской фигуры, ограниченной кривыми
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Неизвестный
04.02.2010, 18:09
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это ответ
Здравствуйте, Верещака Андрей Павлович.
Первая кривая - парабола с уравнением y = x2
Вторая - парабола с уравнением y = 2-x2
Найдем точки пересечения:
x2 = 2-x2 <=> x2 = 1 <=> x = { -1; 1 }
Площадь искомой фигуры можно определить как разность площадей областей под графиком второй параболы, заключенной между линиями x = -1; x = 1; y = 0
и под графиком первой параболы с теми же ограничениями.
Площадь под графиком кривой численно равна определенному интегралу в заданных границах.
Первая площадь равна: -1[$8747$]1(2-x2)dx = -1[$8747$]12dx - -1[$8747$]1x2dx
Вторая площадь равна: -1[$8747$]1x2dx
Т.о., площадь фигуры S равна:
S = -1[$8747$]12dx - -1[$8747$]1x2dx - -1[$8747$]1x2dx = -1[$8747$]12dx - 2-1[$8747$]1x2dx =
2x|-11 - 2x3/3|-11 = 2 - (-2) - (2/3 - (-2/3)) = 4 - 4/3 = 8/3 = 22/3
Ответ: площадь фигуры, ограниченной кривыми x2-y=0 и x2+y-2=0 равна 22/3
Первая кривая - парабола с уравнением y = x2
Вторая - парабола с уравнением y = 2-x2
Найдем точки пересечения:
x2 = 2-x2 <=> x2 = 1 <=> x = { -1; 1 }
Площадь искомой фигуры можно определить как разность площадей областей под графиком второй параболы, заключенной между линиями x = -1; x = 1; y = 0
и под графиком первой параболы с теми же ограничениями.
Площадь под графиком кривой численно равна определенному интегралу в заданных границах.
Первая площадь равна: -1[$8747$]1(2-x2)dx = -1[$8747$]12dx - -1[$8747$]1x2dx
Вторая площадь равна: -1[$8747$]1x2dx
Т.о., площадь фигуры S равна:
S = -1[$8747$]12dx - -1[$8747$]1x2dx - -1[$8747$]1x2dx = -1[$8747$]12dx - 2-1[$8747$]1x2dx =
2x|-11 - 2x3/3|-11 = 2 - (-2) - (2/3 - (-2/3)) = 4 - 4/3 = 8/3 = 22/3
Ответ: площадь фигуры, ограниченной кривыми x2-y=0 и x2+y-2=0 равна 22/3
5
Спасибо
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