A Nonnegative Continuous Function with Zero Integral

In the setting of Single-Variable Calculus on an Interval, let $a,b\in\mathbb{R}$ with $a<b$, let $[a,b]$ be the closed interval determined by $a$ and $b$, and let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$. Then $f$ is Riemann integrable on $[a,b]$, so the integral appearing below is defined. Suppose that

$$0\le f(x)\quad\text{for every }x\in[a,b],\qquad\text{and}\qquad \int_a^b f(t)\,dt=0 .$$

Problem. Show that $f(x)=0$ for every $x\in[a,b]$.