Is the Riemann integral of a strictly positive function positive? In the proof here a strictly positive function in $(0,\pi)$ is integrated over this interval and the integral is claimed as a positive number. It seems intuitively obvious as the area enclosed by a continuous function's graph lying entirely above the x-axis and the x-axis should not be zero. But how can I prove this formally?
If the function is positive over a closed interval apparently the result is not true (link goes to page 147 in Theories of Integration by Kurtz and Swarz). This has further confused me. Can someone please clarify my doubt.
Thanks

Let $f : I \to \mathbb{R}$ be a function on some interval $I$. 
If $f$ is continuous and positive on $I$, then $\displaystyle \int_I f >0$. Indeed, $f \geq \alpha >0$ on some closed interval $K \subset I$, so $\displaystyle \int_I f \geq \int_K f \geq \alpha \cdot \mu(K)>0$.
