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.

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.