Define $[x]$ as the greatest integer less than or equal to $x$, for each $x\in \left (- \infty, \infty \right ).$ If $y = [x]$, then area under $y$ for $x\in \left [ 1,4 \right ]$ is _______.

The graph of $y = [x]$ for $x \in [1, 4]$.

The required area $ = (1 \times 1) + (1 \times 2) + (1 \times 3) = 1 + 2 + 3 = 6 \;\text{unit}^{2}. \quad [\because $ Area of rectangle = Base $ \times $ Height $]$

So, the correct answer is $(D).$