The metric $d(f,g)=\int_a^b|f(x)-g(x)|dx$ appeared twice when I was studying. The author said that the space of Riemann integrable function with the metric $d$ is not complete, but the space $L^1$ yes, it is.
But... There is metrics that makes a lot of function spaces complete. Why is the metric $d$ so special? It has a name?
(apologize the bad English, please)
The map $f \mapsto \int_a^b f(x)\,dx$ is continuous with respect to the metric $d$. So if you have a sequence $f_n \to f$ converging in the metric $d$, you know that $\int_a^b f_n(x)\,dx \to \int_a^b f(x)\,dx$. In other words, convergence in the metric $d$ is exactly the right thing to guarantee that you can pass the limit under the integral sign.