To find the volume of the solid, we need to integrate the area of the cross-sections perpendicular to the $x$-axis. Each cross-section is a square, and the base of each square runs from the semicircle $y = -\sqrt{81 - x^2}$ to the semicircle $y = \sqrt{81 - x^2}$.

Step-by-Step Solution

1. Determine the length of the side of the square:

   The length of the base of each square at a given $x$ is the vertical distance between the two semicircles:

   $\text{Side length of square} = \sqrt{81 - x^2} - (-\sqrt{81 - x^2}) = 2\sqrt{81 - x^2}.$

2. Find the area of each square cross-section:

   Since the side length of each square is $2\sqrt{81 - x^2}$, the area $A(x)$ of each square is:

   $A(x) = \left(2\sqrt{81 - x^2}\right)^2 = 4(81 - x^2) = 324 - 4x^2.$

3. Set up the integral for the volume:

   The volume of the solid can be found by integrating the area of the cross-sections from $x = -9$ to $x = 9$:

   $V = \int_{-9}^{9} A(x) \, dx = \int_{-9}^{9} (324 - 4x^2) \, dx.$

4. Compute the integral:

   We can split the integral and compute it:

   $V = \int_{-9}^{9} 324 \, dx - \int_{-9}^{9} 4x^2 \, dx.$

   Now, compute each integral separately:

   $\int_{-9}^{9} 324 \, dx = 324 \cdot (9 - (-9)) = 324 \cdot 18 = 5832.$

   For the second integral:

   $\int_{-9}^{9} 4x^2 \, dx = 4 \left( \int_{-9}^{9} x^2 \, dx \right).$

   Since $x^2$ is an even function, we can use the property that:

   $\int_{-a}^{a} x^2 \, dx = \frac{2a^3}{3}.$

   Thus,

   $\int_{-9}^{9} x^2 \, dx = \frac{2 \cdot 9^3}{3} = \frac{2 \cdot 729}{3} = 486.$

   Therefore,

   $\int_{-9}^{9} 4x^2 \, dx = 4 \cdot 486 = 1944.$

5. Calculate the volume:

   Now, substitute back into the expression for $V$:

   $V = 5832 - 1944 = 3888.$

Final Answer

The volume of the solid is:

$\boxed{3888}.$

Would you like more details or have any questions? Here are some related questions to explore:

1. What is the effect on volume if the base of the cross-sections changes from a square to a different shape?
2. How would the solution change if the semicircle were defined by a different radius?
3. What would happen if the solid extended between different limits on the $x$-axis?
4. How does the use of symmetry simplify the computation of integrals for solids of revolution?
5. What are some other real-world applications of integrals for finding volumes of solids?

Tip: Always consider the geometric interpretation of your integral to verify your results visually.