Understanding the Problem: Area Bounded by the Greatest Integer Function

The question asks for the area of the region enclosed by the graph of the function $y = \lfloor x \rfloor$, the x-axis (which is the line $y=0$), and the vertical lines $x = -1.5$ and $x = -1.8$. The symbol $\lfloor x \rfloor$ represents the greatest integer function, also known as the floor function. It gives the largest integer less than or equal to $x$.

Analyzing the Greatest Integer Function $\lfloor x \rfloor$ in the Given Interval

The interval specified is from $x = -1.8$ to $x = -1.5$. We need to determine the value of $\lfloor x \rfloor$ for $x$ within this interval.

• For any $x$ such that $-2 \le x < -1$, the greatest integer less than or equal to $x$ is $-2$.
• Our interval is $[-1.8, -1.5]$. Both $-1.8$ and $-1.5$ are between $-2$ and $-1$.
• Specifically, $-2 \le -1.8 < -1$ and $-2 \le -1.5 < -1$.
• Therefore, for every $x$ in the interval $[-1.8, -1.5]$, the value of $\lfloor x \rfloor$ is $-2$.

So, the function $y = \lfloor x \rfloor$ simplifies to the constant function $y = -2$ for all $x$ in the interval $[-1.8, -1.5]$.

Identifying the Bounded Region

The region whose area we need to find is bounded by:

• The line $y = -2$ (from $y = \lfloor x \rfloor$).
• The line $y = 0$ (the x-axis).
• The vertical line $x = -1.8$.
• The vertical line $x = -1.5$.

This description defines a rectangle in the coordinate plane.

Calculating the Area of the Bounded Region

The region is a rectangle with vertices at the points $(-1.8, 0)$, $(-1.5, 0)$, $(-1.5, -2)$, and $(-1.8, -2)$.

To find the area of this rectangle, we need its width and height.

• Width: The distance between the two vertical lines $x = -1.8$ and $x = -1.5$. This is the difference between the x-coordinates:

  \text{Width} = |-1.5 - (-1.8)| = |-1.5 + 1.8| = |0.3| = 0.3

• Height: The distance between the x-axis ($y=0$) and the line $y = -2$. This is the difference between the y-coordinates:

  \text{Height} = |0 - (-2)| = |0 + 2| = |2| = 2

  Note: While the function's value is negative, area is always a positive quantity, so we take the absolute value of the height.

The area of the rectangle is the product of its width and height.

\text{Area} = \text{Width} \times \text{Height} = 0.3 \times 2 = 0.6

Final Answer Calculation Summary

The area bounded by the given curves and lines is 0.6 square unit.

Revision Table: Greatest Integer Function Area

Additional Information: Properties of the Greatest Integer Function

The greatest integer function, $y = \lfloor x \rfloor$, has a graph that looks like a series of steps. It jumps at integer values of $x$.

• For $x \in [n, n+1)$, where $n$ is an integer, $\lfloor x \rfloor = n$.
• Examples: $\lfloor 3.7 \rfloor = 3$, $\lfloor 0.5 \rfloor = 0$, $\lfloor -1.2 \rfloor = -2$, $\lfloor 4 \rfloor = 4$.
• The domain of $\lfloor x \rfloor$ is all real numbers ($\mathbb{R}$).
• The range of $\lfloor x \rfloor$ is the set of all integers ($\mathbb{Z}$).
• Area calculations involving $\lfloor x \rfloor$ over an interval often break down into summing the areas of several rectangles, as the function is constant over integer intervals. However, in this specific problem, the interval $[-1.8, -1.5]$ falls entirely within one such constant interval $[-2, -1)$, simplifying the calculation to a single rectangle.