Помогите, пожалуйста!
Найти объем тела, ограниченного поверхностями z = 2 - x, y^2 = 2 * x, x = 2, z = 0.

z = 2 - x, y^2 = 2 * x, x = 2, z = 0

y^2 = 2 * x, x = 2 => y^2 = 2 * 2 = 4 => y = +- 2.
Тогда
V = int (-2 2) dy int (y^2/2 2) dx int (0 2-x) dz =
= int (-2 2) dy int (y^2/2 2) (z)_{0}^{2 - x} dx = int (-2 2) dy int (y^2/2 2) (2 - x) dx =
= int (-2 2) dy (2 * x - 1/2 * x^2)_{y^2/2}^{2} =
= int (-2 2) dy ((2 * 2 - 1/2 * 2^2) - (2 * y^2/2 - 1/2 * (y^2/2)^2)) =
= int (-2 2) (4 - 2 - y^2 + 1/2 * y^4/4) dy = int (-2 2) (2 - y^2 + 1/8 * y^4) dy =
= (2 * y - 1/3 * y^3 + 1/8 * 1/5 * y^5)_{-2}^{2} =
= (2 * 2 - 1/3 * 2^3 + 1/40 * 2^5) - (2 * (-2) - 1/3 * (-2)^3 + 1/40 * (-2)^5) =
= (4 - 8/3 + 4/5) - (-4 + 8/3 - 4/5) = 4 - 8/3 + 4/5 + 4 - 8/3 + 4/5 =
= 8 - 16/3 + 8/5 = 120/15 - 80/15 + 24/15 = 64/15
Ответ: V = 64/15.