$\displaystyle 0 = \int f$ but $\displaystyle \lim\int f_n = \lim\int_{[0,n]}\frac1n =\lim \frac1n\cdot n = \lim1 = 1$.
Basically, although the functions decay in height, they grow in width by the same amount. Since the domain is infinite, they can grow arbitrarily much in width so this growth can counteract any amount of decrease in height.