Question

三重积分计算∫∫∫ycos(x+z)dv,Ω由y=√x,y=0,z=0,x+z=π/2围成的闭区域,

Answer 1

原式=∫dx∫ydy∫cos(x+z)dz
=∫dx∫y(1-sinx)dy
=(1/2)∫x(1-sinx)dx
=(1/2)[∫xdx-∫xsinxdx]
=(1/2)[π²/8-∫xsinxdx]
=(1/2)(π²/8-1) (应用分部积分法)
=(π²-8)/16.