Let $f:[a,b]\longrightarrow{R}$ continuous, we know that $$\int\_{a}^{b} f^4(x) dx=0$$ Can we say that $f=0$?

Now, I am pretty sure the answer to this question is no. For example, I know that in the function definite in $[0,2]$, 

$f(x) = \begin{cases} 0, &x \neq 0 \\ 1, & x=0 \end{cases}$
it's area is $\int\_{0}^{2} f(x)dx=0$ 

And this gives me an idea on how there are functions which have an area of $0$ even when $f\neq0$