Let $a, b \in \mathbb{R}$ such that $a<b$. For each $n \in \mathbb{N}$, let $f_n : [a,b] \subset \mathbb{R} \longrightarrow \mathbb{R}_+$ be a continuous functions. Since, for each $n \in \mathbb{N}$, the function $f_n$ is continuous, then $f_n$ is integrable. Hence, there exists a constante $M_n >0$ (depending of $n \in \mathbb{N}$) such that $$\int_{a}^{b} f_n(x) \; dx < M_n.\tag{1}$$
Question. There exists $M>0$ (not depending of $n \in \mathbb{N})$ so that $$\int_{a}^{b} f_n(x) \; dx < M, \; \forall \; n \in \mathbb{N}? \tag{2}$$
It seems to me unlikely that happen, since we do not have more properties of functions $ f_n $, with $ n \in \mathbb{N} $, for instante, $ f_n $ is not necessarily bounded.
What about $f_n(x)=n$ for every $x \in [a,b]$ ?
(Note also that your last sentence is not correct : as continuous functions over $[a,b]$, all the $f_n$'s are bounded)