连续非负函数积分等于零则函数恒为零的证明问询

Suppose that $f$ is continuous on $[a, b]$, that $f(x) \ge 0$ for all $x \in [a, b]$ and that $\int_a^b f = 0$. Prove that $f(x) = 0$ for all $x \in [a, b]$.

我的尝试：

Let $\dot{\Pi}$ be a tagged partition of $[a, b]$. We have that for any $\varepsilon > 0$ there is some $\delta_\varepsilon > 0$ such that $|\dot{\Pi}| < \delta_\varepsilon$ and
$|S(f, {\dot{\Pi}})| = |\sum_{i} f(t_i)(x_{i}-x_{i-1})| < \varepsilon$.

Assume $f(t_i) \neq 0$ for some $t_i \in [a, b]$...（原文内容此处截断）

备注：内容来源于stack exchange，提问作者user13