Talking with a friend, he was trying to say that for $f:[a,b]\to\mathbb{R}$ such that $f(x)>0$ for each $x \in [a,b]$ we have $\int_a^b f(x)dx>0$ because, if $F$ is the integral function of $f$ with $a$ as lower bound of integration, it is $F'(x)=f(x)>0$ and so $F$ is increasing hence $\int_a^b f(x)dx=F(b)-F(a)>0$ because $b>a$ and $F$ is increasing.
I don't agree with this solution: we have no hypothesis about the continuity of $$, hence the fundamental theorem of calculus does not hold and so we can't say that $F'(x)=f(x)$ and that $\int_a^b f(x)dx=F(b)-F(a)$. Am I correct or my friend's…