Задача 59445 ...
Условие
∭_(V) 2xy²z dxdydz,
V: 0 ≤ x ≤ 4, -2 ≤ y ≤ 2, 0 ≤ z ≤ x.
математика
263
Решение
★
D: 0 ≤ x ≤ 4; -2 ≤ y ≤ 2
=∫∫_(D) dxdy( [blue]∫_(0) ^(x)2xy^2zdz[/blue])=∫∫_(D) dxdy(2xy^2)*(z^2/2)|_(0)^(x)dxdy=
=∫_(0)^(4)(∫_(-2)^(2)x^3ydy)dx=
=∫_(0)^(4)(x^3)*(y^2/2)|_(-2)^(2)dx=
=∫_(0)^(4)(x^3)*(2^2/2- (-2)^2/2))dx=∫_(0)^(4)(x^3)*4dx=4∫_(0)^(4)(x^3)dx=4*(x^4/4)|_(0)^(4)=4^4=[b]256[/b]