Maybe a trivial question. I think it is true, but I do not succeed in proving.
Suppose $h(x)$ is such that for all $a\geq 0$ $ \int_a^\infty h(x) dx\geq0 $ Prove for all $a\geq 0$ $ \int_a^\infty x h(x) dx\geq0 $
If needed, one may assume $\lim_{x\rightarrow\infty}h(x)=0$ and/or continuity of $h$.