I have to prove the following $$ \left| \int_\Gamma F \cdot d\gamma \right| \leq M\cdot l(\Gamma) $$ where $M= \max\{ \|F(x)\|:x\in \Gamma \}$ and $l(\Gamma)$ is the lenght of $\Gamma$.
I was using the hypothesis that if for every $\gamma(t) \in \Gamma,\ F(\gamma(t))$ is in the same direction as the vector $\gamma'(t)$ then we have that $F(\gamma(t)) = \alpha(t)\gamma'(t) $ with $\alpha(t)\geq 0, \forall t$. And then
$$ \int_a^b \alpha(t)\gamma'(t)\cdot \gamma'(t)\,dt = \int_a^b \alpha(t) \|\gamma'(t)\|^2 \, dt $$ $$ = \int_a^b (\alpha(t)\| \gamma'(t)\|) \|\gamma'(t)\| \, dt = \int_a^b \|F\| \|\gamma'(t)\| \, dt $$
Therefore $$ \int_\Gamma F \cdot d\gamma = \int_\Gamma \|F\|\|d\gamma\|$$
From there is very easy to see that $$ \int_\Gamma F \cdot d\gamma \leq M\cdot l(\Gamma) = M\int_a^b \|\gamma'(t)\| \, dt $$
But I don't really know how to prove it for the general case, do you have any ideas?