原式=∫(0,4)ydy∫(y/2,6-y)e^xdx
=∫(0,4)y[e^(6-y)-e^(y/2)]dy
=[-ye^(6-y)]│(0,4)+∫(0,4)e^(6-y)dy+[-2ye^(y/2)]│(0,4)+2∫(0,4)e^(y/2)dy (应用分部积分法)
=-4e²+[e^(6-y)]│(0,4)-8e²+[4e^(y/2)]│(0,4)
=-4e²+(e^6-e²)-8e²+4(e²-1)
=e^6-9e²-4.