Let $f_n \to f$ uniformly in $[a,b]$. If each $f_n$ is integrable in $[a,b]$,then $f$ is integrable in $[a,b]$ and $\int_a^b f_n(x) dx \to \int_a^b f(x) dx$.
If $\int_a^b f_n(x) dx \nrightarrow \int_a^b f(x) dx$,then the convergence is not uniform.
But, if $\int_a^b f_n(x) dx \to \int_a^b f(x) dx$, can we conclude then that the converge is uniform? Is it a sufficient and necessary condition?