To solve this problem using the general slicing method, we will follow the steps below:

Problem Restatement:

The solid has a semicircular base of radius 5. The cross-sections perpendicular to the base (parallel to the diameter) are squares. We are tasked with finding the volume of this solid.

Step 1: Set Up the Coordinate System

Let’s place the semicircle in the xy-plane. The equation for the upper half of a circle with radius 5, centered at the origin, is:

$y = \sqrt{25 - x^2}$

This describes the top half of the semicircle with its base along the x-axis.

Step 2: Define the Cross-Sections

At any point x along the diameter, the length of the side of the square (the cross-section) is equal to the vertical distance from the x-axis to the semicircle. This distance is given by the equation for $y$, i.e., the side of the square at each point is:

$\text{side length of the square} = 2y = 2\sqrt{25 - x^2}$

This is because the total length of the square’s side spans from the upper edge of the semicircle to the lower edge along the x-axis.

Step 3: Volume of the Slice

The area of the square cross-section is:

$\text{Area of square} = (\text{side length})^2 = \left( 2\sqrt{25 - x^2} \right)^2 = 4(25 - x^2)$

To find the volume of the solid, we integrate this area across the length of the base, from $x = -5$ to $x = 5$.

Step 4: Set Up the Integral

The volume $V$ of the solid is given by the integral of the area of the cross-section along the x-axis:

$V = \int_{-5}^{5} 4(25 - x^2) \, dx$

Step 5: Compute the Integral

First, expand the integrand:

$V = 4 \int_{-5}^{5} (25 - x^2) \, dx$

Now, break this into two simpler integrals:

$V = 4 \left( \int_{-5}^{5} 25 \, dx - \int_{-5}^{5} x^2 \, dx \right)$

1. Integrating $25$:

$\int_{-5}^{5} 25 \, dx = 25x \Big|_{-5}^{5} = 25(5) - 25(-5) = 250$

2. Integrating $x^2$:

Since $x^2$ is an even function, the integral from $-5$ to $5$ is twice the integral from 0 to 5:

$\int_{-5}^{5} x^2 \, dx = 2 \int_{0}^{5} x^2 \, dx = 2 \cdot \left( \frac{x^3}{3} \Big|_0^5 \right) = 2 \cdot \left( \frac{125}{3} \right) = \frac{250}{3}$

Step 6: Final Calculation

Now, substitute these results back into the volume equation:

$V = 4 \left( 250 - \frac{250}{3} \right)$

Simplify:

$V = 4 \left( \frac{750}{3} - \frac{250}{3} \right) = 4 \left( \frac{500}{3} \right) = \frac{2000}{3}$

Final Answer:

The volume of the solid is:

$V = \frac{2000}{3} \, \text{cubic units}$

Would you like to go over the steps again or have any questions? Here are some related topics for further exploration:

1. How does the volume change if the cross-sections were circles instead of squares?
2. Can we use the same method to calculate the volume of solids with triangular cross-sections?
3. What happens to the volume if the radius of the semicircle changes to 10?
4. How do you compute volumes of solids with irregular cross-sections?
5. How would you approach the volume of a solid with elliptical cross-sections?

Tip: Always visualize the problem and sketch the region before setting up integrals, especially when dealing with solids with complex cross-sections.