Question

求二重积分∫(0,2)dx∫(x,2)e^(-y^2)dy

Answer 1

交换积分次序：

∫(0,2)dx∫(x,2)e^(-y²)dy

=∫(0,2)dy∫(0,y)e^(-y²)dx

=∫(0,2) ye^(-y²)dy

=(1/2)∫(0,2) e^(-y²)d(y²)

=(-1/2)e^(-y²) |(0,2)

=(1/2)(1-e⁻⁴)