Let {$f\_n$} be a uniformly bounded sequence of Riemann int'ble functions on $[a,b]$.If $f\_n\rightarrow 0$ pointwise then does it follow that $\int \_{[a,b]}f\_n\rightarrow0$?

My thoughts: The result doesn't follow from the given assumptions. To justify my claim, I choose $f\_n(x)=\frac{x^2}{x^2+(1-nx)^2}$ on $[0,1]$ which satisfies all the criteria. Clearly, $f\_n\rightarrow 0$ pointwise but I haven't been able to show that $\int \_{[a,b]}f\_n$ doesn't converge to $0$ although it's clear that it doesn't.

Are there any other counter-examples to justify this result?. I came up with $f\_n(x)=nx(1-x^2)^n$ on $[0,1]$ but this choice of function doesn't have the uniform boundedness. Can anybody provide me with a relatively easy example to go with?