Question

问一道三重积分的问题

设区域:Ω={（x,y,z)|0<=z<=t,x^2+y^2<=t^2}(t>0),函数f(u)可导并且f(0)=0,f'(0)=2,F(t)=∫∫∫f（x^2+y^2)dxdydz,求limF(t)/t^5 (t趋向于0）
详见：

Answer 1

F(t) = ∫<0,t)dz∫<0,2π>dt∫<0,t>f(r^2)rdr
= πt∫<0,t>f(r^2)dr^2 = πt∫<0,t^2>f(u)du.
则 lim<t→0>F(t)/t^5
= πlim<t→0>∫<0,t^2>f(u)du/t^4 (0/0)
= πlim<t→0>2tf(t^2)/(4t^3)
= πlim<t→0>f(t^2)/(2t^2) (0/0)
= πlim<t→0>2tf'(t^2)/(4t)= πlim<t→0>f'(t^2)/2 = π.