Question

∫(1/4,1/2)dy∫(1/2,√y)e^(y/x)dx+∫(1/2,1)dy∫(y,√y)e^(y/x)dx=?

Answer 1

积分区域是y=x²,y=x,x=1/2围成的区域

交换积分次序

原式=∫(1/2,1)dx∫(x²,x)e^(y/x)dx

=∫(1/2,1) dx ∫(x²,x) d(xe^(y/x)

=∫(1/2,1) x(e-e^x)dx

=∫(1/2,1) d((1/2)ex²-(x-1)e^x)

=(3/8)e-(1/2)e^(1/2)