Задача 70164 ...

636012a2944a02358e896d70

16.03.2023 23:02:26

Решить двойной интеграл (4xy+16x^(3)y^(3))dxdy D: x=1; y=x^(3) y=–∛x

5f3ea7e3faf909182968ddd9

17.03.2023 00:11:15

★

D:
0 ≤ x ≤ 1
-∛ x≤ y ≤ x^3

= ∫^(1) _(0) [blue]∫ ^(x^3)_(-∛x)(4xy+16x^3y^3)dy[/blue])dx=

=∫^(1) _(0) [blue](4x*(y^2/2)+16x^3((y^4/4))| ^(x^3)_(-∛x)[/blue])dx=

=∫^(1) _(0)((4x*((x^3)^2/2)+16x^3(((x^3)^4/4) - (4x*((-∛x)^2/2)+16x^3(((-∛x)^4/4))dx=

=∫^(1) _(0)(2x^7+4x^(15)+2x^(5/3) -4x^(13/3))dx=(2*(x^8/8)+4*(x^(16)/16)+2*x^(8/3)/(8/3)-4*x^(16/3)/(16/3))|^(1)_(0)=

=(2/8)+(4/16))+(6/8)-(12/16)=[b]1/2[/b]