Здравствуйте
Помогите, пожалуйста, вычислить объем тела, ограниченного поверхностями
x = 0, z = 0, x + y = 4, z = 4 * y^(1/2).

x = 0, z = 0, x + y = 4, z = 4 * y^(1/2) => y >= 0.
Пределами интегрирования будут:
0 <= z <= 4 * y^(1/2)
0 <= y <= 4 - x
0 <= x <= 4
Получаем, что
V = int (0 4) dx int (0 4 - x) 4 * y^(1/2) dy =
= int (0 4) dx (4 * 1/(1/2 + 1) * y^(1/2 + 1))_{0}^{4 - x} =
= int (0 4) dx (4 * 2/3 * y^(3/2))_{0}^{4 - x} =
= int (0 4) dx (8/3 * (4 - x)^(3/2) - 8/3 * 0^(3/2)) =
= int (0 4) 8/3 * (4 - x)^(3/2) dx = 8/3 * int (0 4) (4 - x)^(3/2) dx =
= | 4 - x = t, x = 4 - t, dx = -dt | = -8/3 * int (4 0) t^(3/2) dt =
= 8/3 * int (0 4) t^(3/2) = 8/3 * (1/(3/2 + 1) * t^(3/2 + 1))_{0}^{4} =
= 8/3 * (2/5 * t^(5/2))_{0}^{4} = 8/3 * (2/5 * 4^(5/2) - 2/5 * 0^(5/2)) =
= 8/3 * 2/5 * 32 = 16/15 * 32 = 512/15
Ответ: V = 512/15.