Constructing a Sequence for Pointwise Convergence and Unboundedness

[Arkuski]

Give an example of a sequence $\{ f_n\}$ of continuous functions defined on [0,1] such that $\{ f_n\}$ converges pointwise to the zero function on [0,1], but the sequence $\{ \int^{1}_{0} f_n\}$ is unbounded.

I'm pretty lost on this one.

 

[jbunniii]

Hint: Try coming up with a sequence of functions such that $f_n(x)$ is only nonzero on the interval $(0, 1/n)$. You will need to make them grow "taller" and "narrower" as $n$ increases.