define $\displaystyle f:[0,6] \to \mathbb{R} $
$\displaystyle f(x)=\left\{\begin{array}{ll}x&\;\;if\,\,x \ is \ rational\\0&\;\; \ x \ is \ irrational \end{array}\right.$
prove that f is not integrable over [0,6]
Exact same argument as you have seen before. It suffices to prove that isn't integrable on $\displaystyle [0,1]$ then given any partition $\displaystyle \left\{x_0,\cdots,x_n\right\}$ then $\displaystyle \sup_{x\in[x_{j-1},x_j]}f(x)=x_j>0,\inf_{x\in[x_{j-1},x_j]}f(x)=0$