高數打卡05

$設函數f(x)連續且恆大於零，$
$\begin{aligned} &F(t)=\frac{\iiint_{\Omega(t)} f\left(x^{2}+y^{2}+z^{2}\right) d v}{\iiint_{D(t)} f\left(x^{2}+y^{2}\right) d \sigma}\\ \end{aligned}$
$\begin{aligned} G(t)=\frac{\iint_{D(t)} f\left(x^{2}+y^{2}\right) d \sigma}{\int_{-t}^{t} f\left(x^{2}\right) d x} \end{aligned}$
$其中\Omega(t)=\{(x,y,z)|x^2+y^2+z^2 \leq t^2\},D(t)=\{(x,y)|x^2+y^2 \leq t^2\}.$
$(1)討論F(t)在區間(0,+\infty)內的 單調性；$
$(2)證明當t>0時，F(t)>\frac{2}{\pi}G(t).$