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To solve this problem, let's break it down step-by-step.

1. **Identify the cross-sectional shape and dimensions**:
   - The problem states that the cross-sections perpendicular to the \( x \)-axis are squares.
   - The diagonals of these squares run from the parabola \( y = -2\sqrt{x} \) to \( y = 2\sqrt{x} \).

2. **Determine the length of the diagonal**:
   - The distance between \( y = -2\sqrt{x} \) and \( y = 2\sqrt{x} \) is the length of the diagonal of the square.
   - The length of the diagonal, \( d \), is the difference between the two functions:
     \[
     d = 2\sqrt{x} - (-2\sqrt{x}) = 4\sqrt{x}
     \]

3. **Calculate the side length of the square**:
   - For a square with a diagonal \( d \), the side length \( s \) is given by \( s = \frac{d}{\sqrt{2}} \).
   - Substitute \( d = 4\sqrt{x} \):
     \[
     s = \frac{4\sqrt{x}}{\sqrt{2}} = 2\sqrt{2} \sqrt{x} = 2\sqrt{2x}
     \]

4. **Find the area of each square cross-section**:
   - The area \( A \) of a square with side length \( s \) is \( s^2 \).
   - Substitute \( s = 2\sqrt{2x} \):
     \[
     A = (2\sqrt{2x})^2 = 4 \cdot 2x = 8x
     \]

5. **Set up the integral for the volume**:
   - The volume of the solid is found by integrating the area of the cross-section along the interval \( x = 0 \) to \( x = 18 \):
     \[
     V = \int_0^{18} A \, dx = \int_0^{18} 8x \, dx
     \]

6. **Evaluate the integral**:
   \[
   V = \int_0^{18} 8x \, dx = 8 \int_0^{18} x \, dx
   \]
   Using the integral \( \int x \, dx = \frac{x^2}{2} \):
   \[
   V = 8 \left[ \frac{x^2}{2} \right]_0^{18} = 8 \cdot \frac{18^2}{2} = 8 \cdot \frac{324}{2} = 8 \cdot 162 = 1296
   \]

Therefore, the volume of the solid is **1296 cubic units**.

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Would you like more details on any step?

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Here are some related questions to deepen your understanding:

1. How do we derive the relationship between the diagonal and side length of a square?
2. What would happen to the volume if the cross-sections were circles instead of squares?
3. Can we find the volume if the functions defining the diagonals were different?
4. What if the bounds of integration changed from 0 to 18 to 0 to some other value?
5. How would the setup change if the cross-sectional shape was an equilateral triangle?

**Tip:** When dealing with solids of known cross-section, focus on finding the area of the cross-section as a function of the variable along the axis of revolution, then integrate it over the given interval.

A solid lies between planes perpendicular to the x-axis at x = 0 and x = 18. The cross-sections perpendicular to the axis on the interval 0 ≤ x ≤ 18 are squares with diagonals that run from the parabola y = -2√x to the parabola y = 2√x. Find the volume of the solid.

Calculate the volume of a solid with square cross-sections on the interval 0 ≤ x ≤ 18, where diagonals run from the parabola y = -2√x to y = 2√x. This problem involves integral calculus and geometry of squares and parabolas, making it suitable for undergraduate calculus students.

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