Задача 56604 ...

5fe86c9a1c6d1142ab519132

28.12.2020 07:23:20

Вычислить: ∬_(D) y^2 e^(-xy/4) dx dy , D: y = 2 , y = x , x = 0.

5f3ea7e3faf909182968ddd9

28.12.2020 21:49:14

★

= ∫ ^(1)_(0)[blue][b]( ∫ ^(x=y)_(x=0)y^2e^(-xy/4)dx)[/b][/blue]dy=∫ ^(1)_(0)[blue][b]( -4/y)∫ ^(x=y)_(x=0)y^2e^(-xy/4)d(-xy/4)[/b][/blue]dy=

=∫ ^(1)_(0)[blue][b]( -4y^2/y)*(e^(-xy/4 ))|^(x=y)_(x=0)[/b][/blue]dy=-4∫ ^(1)_(0)[blue][b]y*(e^(-y*y/4 )-e^(0))[/b][/blue]dy=


=-4∫ ^(1)_(0)(y*e^(-y^2/4)-y)dy=(-4*(-2)e^(-y^2/4)+4*(y^2/2))|^(1)_(0)=8e+2-(8+0)=[b]8e-6[/b]