We are tasked with finding the volume of a solid whose cross-sections perpendicular to the $x$-axis are squares. The diagonal of each square runs from the parabola $y = -2\sqrt{x}$ to $y = 2\sqrt{x}$, for $x$ ranging from $0$ to $12$.

Step 1: Determine the length of the diagonal of each square

The length of the diagonal at a given $x$-value is the vertical distance between the two parabolas. This is calculated as:

$\text{Diagonal} = 2\sqrt{x} - (-2\sqrt{x}) = 4\sqrt{x}$

Step 2: Relating the diagonal to the side length of the square

Let $s(x)$ represent the side length of the square at a given $x$-value. Since the diagonal $d(x)$ and the side length of a square are related by the formula $d = s\sqrt{2}$, we can express $s(x)$ as:

$s(x) = \frac{d(x)}{\sqrt{2}} = \frac{4\sqrt{x}}{\sqrt{2}} = \frac{4}{\sqrt{2}}\sqrt{x} = 2\sqrt{2}\sqrt{x}$

Step 3: Calculate the area of the square cross-section

The area $A(x)$ of each square cross-section is given by the square of the side length:

$A(x) = s(x)^2 = \left(2\sqrt{2}\sqrt{x}\right)^2 = 8x$

Step 4: Set up the volume integral

The volume of the solid is found by integrating the area of the cross-sections from $x = 0$ to $x = 12$:

$V = \int_0^{12} A(x) \, dx = \int_0^{12} 8x \, dx$

Step 5: Evaluate the integral

Now, we can evaluate the integral:

[ V = 8 \int_0^{12} x , dx = 8 \left[\frac{x^2}{2}\right]_0^{12} = 8 \left[\frac{12^2}{2} - \frac{0^2}{2}\right] ] $V = 8 \times \frac{144}{2} = 8 \times 72 = 576$

Final Answer:

The volume of the solid is $576$ cubic units.

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Would you like further details or have any questions? Here are some related problems you can explore:

1. What if the cross-sections were equilateral triangles instead of squares?
2. How does the volume change if the parabolas are shifted vertically?
3. What is the volume if the diagonal of the square is instead proportional to $x^2$?
4. What would the integral look like if the cross-sections were semicircles?
5. How do we find the centroid of the solid in this case?

Tip: For volume problems involving cross-sectional areas, always start by relating the geometric property (side, diagonal, etc.) to the variable $x$.