计算二重积分：∫[0,1]dx∫[0,x^½]e^(-y²/2)dy

题目

计算二重积分：∫[0,1]dx∫[0,x^½]e^(-y²/2)dy

答案

原式=∫dy∫e^(-y²/2)dx (作积分顺序变换)=∫(1-y²)e^(-y²/2)dy=∫e^(-y²/2)dy-∫y²e^(-y²/2)dy=∫e^(-y²/2)dy-{[-ye^(-y²/2)]│+∫e^(-y²/2)dy} (应用分部积分法)=∫...