Question

计算I=∫(0,1)dx∫(0,1-x)dy∫(x+y,1)sinz/zdz

Answer 1

∫(0,1)dx∫(0,1-x)dy∫(x+y,1)sinz/zdz

=∫(0,1)(sinz/z)dz∫(0,z)dy∫(0,z-y)dx (变换积分顺序：把积分区域投影到yoz平面进行积分计算)

=∫(0,1)(sinz/z)dz∫(0,z)(z-y)dy

=∫(0,1)(sinz/z)(z²/2)dz

=(1/2)∫(0,1)zsinzdz

=(1/2)(sin1-cos1) (应用分部积分法计算)

=(sin1-cos1)/2.