重积分高数F（t)=∫∫∫f(x^2+y^2+z^2)dxdydz.区域为X^2+y^2+z^2

Question

重积分高数
F（t)=∫∫∫f(x^2+y^2+z^2)dxdydz.
区域为X^2+y^2+z^2

Answer 1

用球面坐标：
F(t)=∫∫∫f(x^2+y^2+z^2)dxdydz.
=∫(0, 2π)dθ∫(0,π)dφ∫(0,t)r^2sinφf(r^2)dr
=4π∫(0,t)r^2f(r^2)dr
limF(t)/t^5
=limF'(t)/5t^4=lim4πt^2f(t^2)/5t^4=lim4πf(t^2)/5t^2
=(4π/5)lim[2tf'(t^2)]/2t
=(8π/5)limf'(t^2)
=8π/5