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.

Would you like more details on any step?

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.