Question

If the function is positive then its integral is also positive

Answer 1

In the proof here a strictly positive function in (0,π)(0,π) 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 Integrationby Kurtz and Swarz). This has further confused me. Can someone please clarify my doubt.

Thanks