$$\begin{gathered} \left[2x\right]=0\Rightarrow 2x\in [0,1) \\ \Rightarrow x\in [0,1/2)\Rightarrow \left[y\right]=5\Rightarrow y\in [5,6) \end{gathered}$$

Similarly we can consider  $\left[2x\right]=1,2,3,4and5$
 
Form the graph, area is 3 sq.units