求积分∫0→1 [根号下(4-x^2)]dx

来源:学生作业帮助网 编辑:六六作业网 时间:2024/12/26 15:52:45
求积分∫0→1[根号下(4-x^2)]dx求积分∫0→1[根号下(4-x^2)]dx求积分∫0→1[根号下(4-x^2)]dx令x=2sint,t∈[0,π/2],则√(4-x²)=√(4-

求积分∫0→1 [根号下(4-x^2)]dx
求积分∫0→1 [根号下(4-x^2)]dx

求积分∫0→1 [根号下(4-x^2)]dx
令x=2sint,t∈[0,π/2],则
√(4-x²)=√(4-4sin²t)=2cost
dx=2costdt
∴∫(0→1) √(4-x²)dx
=∫(0→π/6) 4cos²tdt
=∫(0→π/6) 2(cos2t+1) dt
=sin2t+2t|(0→π/6)
=√3/2+π/3