Question

高数题，三重积分，求大神

高数题，三重积分，求大神如图

Answer 1

由积分区域为圆柱可知，设x=r*cosθ,y=r*sinθ,可得F(t)= ∫(0,t)dz∫(0,2π)dθ∫(0,t)f(r)rdr
F(t)=2πt*∫(0,t)f(r)rdr, F(0)=0,洛必达法则可得limF(t)/t^4=limF'(t)/4t^3
f(0)=0,可得F'(t)=2π∫(0,t)f(r)rdr+2πt^2*f(t),F'(0)=0,f(0)=0,f'(1)=1洛必达法则可得
limF(t)/t^4=limF''(t)/12t^2=lim[6πt*f(t)+2πt^2*f'(t)]/12t^2
=lim[3πtf(t)+πt^2*f'(t)]/6t^2=lim[3πf(t)+πt*f'(t)]/6t
f(0)=0,f'(1)=1，洛必达则可得
limF(t)/t^4=lim[3πf(t)+πt*f'(t)]/6t=lim[4πf'(t)+πt*f''(t)]/6=2π/3