From your previous question, I'm assuming you have that $f$ is the pointwise limit of the $f_n$; so, $f$ is identically $0$ on $[0,2]$.
$\int_0^2 f_n(x)\, dx$ is indeed $2/3$ and $\int_0^2f(x)\, dx$ is indeed $0$.
Everything is ok, though. There is nothing here assuring that the integrals $\int_0^2 f_n(x)\,dx$ should converge to $\int_0^1 f(x)\,dx $.
In particular, the $f_n$ do not converge uniformly to $f $, as the answer to the aforementioned question shows. You are not guaranteed that $\int_0^2 f_n (x)\,dx \rightarrow \int_0^2 f(x)\,dx$, if you do not have uniform convergence.
I presume the purpose of this exercise is to show that the hypothesis of uniform convergence is needed in the following theorem:
If $(f_n)$ is a sequence of Riemann integrable functions over $[a,b]$ and if $(f_n)$ converges uniformly to the function $f$ on $[a,b]$, then $f$ is Riemann integrable over $[a,b]$ and $\lim\limits_{n\rightarrow\infty }\int_a^b f_n(x)\,dx=\int_a^b f(x)\,dx$.