when does $h(x)^Tx>0$ result in $\int_{0}^x h(y)^Tdy>0$?

Question

Let $h(x)\in R^n$, and $x\in R^n$ satisfy $$h(x)^Tx>0,~~\forall x\neq 0$$ and $$h(x)^Tx=0,~~~\text{when}~~~ x=0$$

is it true that $\int_{0}^x h(y)^Tdy>0$ for all $x\neq 0$?

comment: when n=1, the answer is yes.