To find the volume of the solid with a semicircular base of radius 1, where cross-sections perpendicular to the base and parallel to the diameter are squares, we'll use the general slicing method.

Steps:

1. Understanding the base: The base of the solid is a semicircle of radius 1. The equation of the semicircle can be expressed as: $y = \sqrt{1 - x^2}, \quad -1 \leq x \leq 1$ This represents the upper half of a circle with radius 1, centered at the origin.

2. Square cross-sections: For each value of $x$, the cross-section perpendicular to the x-axis is a square. The side length of this square is equal to the vertical distance between the top of the semicircle and the x-axis, which is $2y$. Therefore, the side length of the square at a given $x$ is: $s(x) = 2y = 2\sqrt{1 - x^2}$

3. Volume of a thin slice: The area $A(x)$ of the square at position $x$ is: $A(x) = s(x)^2 = \left(2\sqrt{1 - x^2}\right)^2 = 4(1 - x^2)$

4. Integrating to find the volume: The volume is obtained by integrating the area of the cross-sections along the x-axis, from $x = -1$ to $x = 1$: $V = \int_{-1}^{1} A(x) \, dx = \int_{-1}^{1} 4(1 - x^2) \, dx$

5. Evaluating the integral: Let's break it down: $V = 4 \int_{-1}^{1} (1 - x^2) \, dx$ We can split the integral: $V = 4 \left( \int_{-1}^{1} 1 \, dx - \int_{-1}^{1} x^2 \, dx \right)$ First, solve each integral: $\int_{-1}^{1} 1 \, dx = 2, \quad \int_{-1}^{1} x^2 \, dx = \frac{2}{3}$

6. Final Calculation: Now substitute these values into the volume equation: $V = 4 \left( 2 - \frac{2}{3} \right) = 4 \left( \frac{6}{3} - \frac{2}{3} \right) = 4 \times \frac{4}{3} = \frac{16}{3}$

Thus, the volume of the solid is:

$V = \frac{16}{3} \text{ cubic units}.$

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Would you like further details or clarifications on any step?

Here are 5 related questions:

1. How would the volume change if the cross-sections were equilateral triangles instead of squares?
2. What is the method for finding the volume of a solid with a circular base and triangular cross-sections?
3. Can you explain how the formula for the area of a semicircle is derived?
4. How does the slicing method apply to finding the surface area of solids?
5. What would happen to the volume if the radius of the semicircle were increased to 2?

Tip: Always visualize the shape and cross-sections to understand how to set up your integrals more effectively!