So the sequence $(f_n)$ is defined over some interval $[a,b]$ and every $f_n$ is continuous (or every $f_n$ is Riemann integrable) and $\lim_{n \to \infty} f_n$ exists pointwise and it equals $f$.
I am trying to understand can in this situation this happen and why:
$$\lim_{n \to \infty} \int_{a}^{b} f_n(x)dx \neq \int_{a}^{b}f(x)dx$$
This can happen if either one of the integrals do not exist or if they both exist but are unequal.
Can someone describe can this happen with the above written assumptions?
The integrals are in the sense of Riemann.
If you don't assume uniform convergence, there are loads of things that can go wrong here.
First of all, if you don't assume continuity of the $f_n$'s, then you can easily imagine a sequence of functions that converges pointwise to the Dirichlet function $$ f(x) = 1_{\mathbb{Q}} $$ For example, take an enumeration of $\mathbb{Q} \cap [a,b]$ and switch on one more point at each step $n$.
You could also consider the sequence $f_n = n \cdot 1_{(0,1/n)}$ which converges pointwise to $0$, but has constant integral.
I'm sure you can also come up with all kinds of nasty stuff, even if you require continuity of the $f_n$'s.