The problem involves computing the volume of an object whose cross-sections, perpendicular to the base, are squares. The base is given as a two-dimensional shape in the xy-plane.

Step-by-Step Approach:

1. Understanding the Base:
   The base is a polygon formed by the points $(1, 1)$, $(3, 1)$, $(4, 0)$, and $(1, -1)$.

2. Equation for Cross-Sectional Area:
   For each value of $x$, the side length of the square is determined by the distance between the top and bottom edges of the shape at that particular $x$-value (the vertical height of the shape at $x$).

3. Finding the Height of the Base Shape:

   • For $x \in [1, 3]$, the top boundary is $y = 1$, and the bottom boundary is a linear function connecting $(1, -1)$ and $(3, 1)$, which has slope 1 and equation $y = x - 2$.
   • For $x \in [3, 4]$, the top boundary is $y = 1$, and the bottom boundary is a horizontal line $y = 0$.

4. Volume Formula: The volume is calculated by integrating the area of the square cross-sections over the $x$-range from 1 to 4. The side length of each square is the vertical distance between the top and bottom boundaries, and the area of each square is the square of this distance. Hence, the volume is given by:

   $V = \int_{1}^{4} \left( \text{side length at } x \right)^2 \, dx$

   We'll split this into two integrals because the base changes at $x = 3$.

   For $1 \leq x \leq 3$: The side length is the distance between $y = 1$ and $y = x - 2$, which is $1 - (x - 2) = 3 - x$.

   For $3 \leq x \leq 4$: The side length is the distance between $y = 1$ and $y = 0$, which is $1 - 0 = 1$.

   Therefore, the volume is:

   $V = \int_{1}^{3} (3 - x)^2 \, dx + \int_{3}^{4} (1)^2 \, dx$

Let me calculate the exact value of this volume.The exact volume of the object is $\frac{11}{3}$, or approximately $3.67$ cubic units when rounded to two decimal places.

Would you like further details on any part of the process? Here are some additional questions you could explore:

1. How do you find the area of cross sections for other shapes, such as semicircles or triangles?
2. What if the base of the object was a different shape? How would the approach change?
3. Can we calculate the centroid of the base shape shown in the graph?
4. How would you calculate the surface area of this 3D object?
5. How would the volume change if the cross sections were not squares but equilateral triangles?

Tip: When dealing with volumes of solids with known cross-sections, always focus on the relationship between the base and the height to set up the correct integral!