Define [x] as the greatest integer less than or equal to x, for each x ϵ (-∞, ∞). If y = [x], then area under y for x ϵ [1,4] is

The correct answer is 6.

Step-by-step explanation:

We want to find the area under the step function defined by:

$y=\left[x\right]$

over the interval:

$x\in [1,4]$

where $\left[x\right]$ is the greatest integer less than or equal to $x$.

By definition, the function $y=\left[x\right]$ takes constant integer values on sub-intervals as follows:

- For $x\in [1,2)$, the value is $y=1$

- For $x\in [2,3)$, the value is $y=2$

- For $x\in [3,4)$, the value is $y=3$

As depicted in the provided graph, the area under the curve is composed of three distinct shaded rectangles labeled 1, 2, and 3. The height of each rectangle corresponds to the constant value of $y$ in that interval, and the width of each interval is $1$:

- Rectangle 1: Width = $2-1=1$, Height = $1$

- Rectangle 2: Width = $3-2=1$, Height = $2$

- Rectangle 3: Width = $4-3=1$, Height = $3$

Therefore, we can calculate the total area as the sum of the areas of these three rectangles:

$$\text{Area}=(1\times 1)+(1\times 2)+(1\times 3)$$

$$\text{Area}=1+2+3=6$$