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The image describes a solid with a circular disk base and parallel cross-sections perpendicular to the base that are squares. The problem asks to:

1. Set up an integral to find the volume \( V \) of the solid.
2. Calculate the volume.

### Step 1: Understanding the Geometry
- The base of the solid is a circular disk with radius \( r \).
- The cross-sections are squares perpendicular to the base.

### Setting Up the Integral
The area of a square depends on the length of its side, which will vary based on its position on the \( x \)-axis. For a circular base, the equation of the circle is:
\[
x^2 + y^2 = r^2
\]
For any cross-section, the side of the square is twice the value of \( y \), which is the distance from the center of the base to the edge of the circle. So, \( y = \sqrt{r^2 - x^2} \), and the side of the square is:
\[
\text{Side length of square} = 2y = 2\sqrt{r^2 - x^2}
\]

The area of a square cross-section is:
\[
\text{Area} = \left(2\sqrt{r^2 - x^2}\right)^2 = 4(r^2 - x^2)
\]

### Volume Integral
To find the volume, we integrate the area of the cross-sections along the \( x \)-axis from \( -r \) to \( r \):
\[
V = \int_{-r}^{r} 4(r^2 - x^2) \, dx
\]

### Step 2: Solving the Integral
We simplify and solve the integral:
\[
V = 4 \int_{-r}^{r} (r^2 - x^2) \, dx
\]
This can be split into two parts:
\[
V = 4 \left[ \int_{-r}^{r} r^2 \, dx - \int_{-r}^{r} x^2 \, dx \right]
\]

#### First Integral:
\[
\int_{-r}^{r} r^2 \, dx = r^2 \int_{-r}^{r} dx = r^2 [x]_{-r}^{r} = r^2(2r) = 2r^3
\]

#### Second Integral:
\[
\int_{-r}^{r} x^2 \, dx = \left[ \frac{x^3}{3} \right]_{-r}^{r} = \frac{r^3}{3} - \left( \frac{(-r)^3}{3} \right) = \frac{2r^3}{3}
\]

Now, substitute both integrals:
\[
V = 4 \left( 2r^3 - \frac{2r^3}{3} \right) = 4 \left( \frac{6r^3}{3} - \frac{2r^3}{3} \right) = 4 \times \frac{4r^3}{3} = \frac{16r^3}{3}
\]

### Final Answer:
The volume of the solid is:
\[
V = \frac{16r^3}{3}
\]

Would you like more details on any part of this process?

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### Related Questions:
1. How is the area of the square related to the equation of the circle?
2. What changes if the cross-sections are triangles instead of squares?
3. Can you set up a similar volume integral for a different 3D solid, like a cone?
4. How does symmetry help in solving these types of problems?
5. What are the limits of integration for a solid with a non-circular base?

### Tip:
When setting up volume integrals, always relate the geometry of the cross-section to the equation of the base!

Consider the following solid S. The base of S is a circular disk with radius r. Parallel cross-sections perpendicular to the base are squares. Set up an integral that can be used to determine the volume V of the solid. Find the volume V of the solid.

Learn how to calculate the volume of a solid with a circular base and square cross-sections using integration. The problem involves setting up and solving a definite integral with bounds from -r to r. Suitable for university-level calculus students.

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