Question

三重积分的计算,第六题,重谢!

Answer 1

原式=∫[0,1]dy∫[0,1-y](1-y)e^[-(1-y-z)^2]dz∫[0,1-y-z]dx

=∫[0,1]dy∫[0,1-y](1-y)(1-y-z)e^[-(1-y-z)^2]dz

=1/2∫[0,1]dy∫[0,1-y](1-y)de^[-(1-y-z)^2]

=1/2∫[0,1](1-y)dy-1/2∫[0,1](1-y)e^[-(1-y)^2]dy

=1/2-1/4-1/4+1/(4e)

=1/(4e)

注：此题的积分区域是一个直四面体,关键是选好积分次序.否则很难做出.