I have a function $f_n:[0,2]->\mathbb R$
$\ f_n(x) = \begin{cases} n^3x^2 & 0 I need to calculate $\int_0^2 f(x)dx$ and $\int_0^2f_n(x)dx$. So $\int_0^2f_n(x)dx=\frac{2}{3}$, but regarding $\int_0^2 f(x)dx$, isnt the integral $\int_0^2 f(x)dx=0?$ If so, why do I get 2 different answers? Im sure $\int_0^2f_n(x)dx=\frac{2}{3}$ is correct.