求积分r^2*(1-r^2)^1/2

Question

求积分 r^2*(1-r^2)^1/2

Answer 1

令r=cosx,x在[0,π],则
∫ r^2*(1-r^2)^1/2dr = ∫(cosx)^2 sinx (-sinx)dx ---积分上下限变为0到π
=-1/4∫(sin2x)^2dx= -1/8∫(1-cos4x)dx=-π/8 +(1/32)∫cos4xd4x = -π/8 +(1/32) sin4x|(0,π) =-π/8