设 $f(x), g(x)$ 在 $[a, b]$ 上连续，且满足
$$
\begin{aligned}
& \int_a^x f(t) \mathrm{d} t \geq \int_a^x g(t) \mathrm{d} t, x \in[a, b) \\
& \int_a^b f(t) \mathrm{d} t=\int_a^b g(t) \mathrm{d} t
\end{aligned}
$$
证明: $\int_a^b x f(x) \mathrm{d} x \leq \int_a^b x g(x) \mathrm{d} x$.