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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.

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    Is $f$ the limit of the $f_n$'s? If so, how are you defining convergence? point-wise? Next, if $f_n \to f$, why should you expect $\int_0^2 f_n \to \int_0^2 f$?2012-02-15
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    You forgot to define $f$. If I guess that $f=0$, you are wondering why limits don't pass through integration I guess. It is because they don't, as this example shows. But under extra hypotheses they do. E.g. see http://en.wikipedia.org/wiki/Dominated_convergence_theorem2012-02-15
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    thanks, yes I was defining it as pointwise.2012-02-15
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    @Frank: Will you please edit your question to say what $f$ is, and clarify that you are asking why you can't pass limits through integrals? The question doesn't make sense as written.2012-02-15

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