Integral of a function in the 1st/3rd quadrant

Question

If $f(x)\cdot x>0$ and $f(0)=0$ it is right that $\int_0^x f(x)dx>0$?

Answer 1

No. Consider $x = 0$.

Assuming $x \neq 0$, consider the two cases of $x > 0$ and $x < 0$. Use the fact that the integral of a Riemann integrable function which is strictly positive (apart from a point) has a strictly positive integral. (See this for example.)
(The answer is "Yes" then.)