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The problem asks for the volume of a solid \( S \) whose base is a semicircular region in the \( xy \)-plane, bounded by \( y = 0 \) and \( y = \sqrt{4 - x^2} \), which is the equation of a semicircle with radius 2. The cross-sections of the solid, perpendicular to the \( x \)-axis, are squares.

### Solution:

1. **Base Region:**
   The base of the solid is the semicircular region, which is given by:
   \[
   y = \sqrt{4 - x^2}, \quad \text{with } -2 \leq x \leq 2
   \]
   This defines a semicircle of radius 2 centered at the origin.

2. **Cross-Sections:**
   The cross-sections perpendicular to the \( x \)-axis are squares. The side length of each square is the distance from \( y = -\sqrt{4 - x^2} \) to \( y = \sqrt{4 - x^2} \), which is simply:
   \[
   \text{Side length of square} = 2\sqrt{4 - x^2}
   \]

3. **Area of Cross-Section:**
   The area of each square is the square of the side length:
   \[
   \text{Area of cross-section} = \left(2\sqrt{4 - x^2}\right)^2 = 4(4 - x^2) = 16 - 4x^2
   \]

4. **Volume of Solid:**
   To find the volume of the solid, integrate the area of the cross-sections along the \( x \)-axis from \( -2 \) to \( 2 \):
   \[
   V = \int_{-2}^{2} (16 - 4x^2) \, dx
   \]

5. **Calculate the Integral:**

   First, split the integral:
   \[
   V = \int_{-2}^{2} 16 \, dx - \int_{-2}^{2} 4x^2 \, dx
   \]
   The first integral is straightforward:
   \[
   \int_{-2}^{2} 16 \, dx = 16[x]_{-2}^{2} = 16(2 - (-2)) = 16 \times 4 = 64
   \]
   The second integral is:
   \[
   \int_{-2}^{2} 4x^2 \, dx = 4 \int_{-2}^{2} x^2 \, dx = 4 \left[\frac{x^3}{3}\right]_{-2}^{2} = 4\left(\frac{8}{3} - \frac{-8}{3}\right) = 4 \times \frac{16}{3} = \frac{64}{3}
   \]

6. **Final Volume:**
   Now subtract the two results:
   \[
   V = 64 - \frac{64}{3} = \frac{192}{3} - \frac{64}{3} = \frac{128}{3}
   \]
   So, the volume of the solid is:
   \[
   V = \frac{128}{3} \, \text{cubic units}
   \]

### Final Answer:
The volume of the solid is \( \frac{128}{3} \) cubic units.

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Would you like any more details on this solution or have any questions?

Here are 5 related questions to expand the concept:

1. What is the formula for the volume of solids with different cross-sectional shapes, such as triangles or rectangles?
2. How do we modify the volume calculation if the cross-sections are not squares but equilateral triangles?
3. Can you find the volume of a solid with a circular base and rectangular cross-sections?
4. How would you approach the problem if the base was a quarter-circle instead of a semicircle?
5. What would change if the cross-sectional shapes were isosceles right triangles instead of squares?

### Tip:
When working with volumes of solids, always carefully determine the area of the cross-section and make sure to properly set the limits of integration.

The base of a solid S is the semicircular region in the xy-plane that is bounded by y = 0 and y = √(4 - x²). If cross sections of S that are perpendicular to the x-axis are squares, then what is the volume of S?

This problem involves calculating the volume of a solid with a semicircular base and square cross-sections perpendicular to the x-axis. The integral is set up to find the volume by integrating the area of the cross-sections from x = -2 to x = 2.

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