高数打卡05

$设函数f(x)连续且恒大于零，$设函数f(x)连续且恒大于零，
$\begin{aligned} &F(t)=\frac{\iiint_{\Omega(t)} f\left(x^{2}+y^{2}+z^{2}\right) d v}{\iiint_{D(t)} f\left(x^{2}+y^{2}\right) d \sigma}\\ \end{aligned}$​F(t)=∭D(t)​f(x2+y2)dσ∭Ω(t)​f(x2+y2+z2)dv​​
$\begin{aligned} G(t)=\frac{\iint_{D(t)} f\left(x^{2}+y^{2}\right) d \sigma}{\int_{-t}^{t} f\left(x^{2}\right) d x} \end{aligned}$G(t)=∫−tt​f(x2)dx∬D(t)​f(x2+y2)dσ​​
$其中\Omega(t)=\{(x,y,z)|x^2+y^2+z^2 \leq t^2\},D(t)=\{(x,y)|x^2+y^2 \leq t^2\}.$其中Ω(t)={(x,y,z)∣x2+y2+z2≤t2},D(t)={(x,y)∣x2+y2≤t2}.
$(1)讨论F(t)在区间(0,+\infty)内的 单调性；$(1)讨论F(t)在区间(0,+∞)内的单调性；
$(2)证明当t>0时，F(t)>\frac{2}{\pi}G(t).$(2)证明当t>0时，F(t)>π2​G(t).