To find the volume of a solid with a semicircular base of radius 11, whose cross-sections perpendicular to the base and parallel to the diameter are squares, we can use the following method:

Step-by-Step Approach:

1. Equation of the semicircle:
   Since the base is a semicircle, the equation of the semicircle (with radius $11$) in the $xy$-plane is: $y = \sqrt{11^2 - x^2} = \sqrt{121 - x^2}$ where $x$ ranges from $-11$ to $11$.

2. Area of the cross-section:
   The cross-sections perpendicular to the base and parallel to the diameter are squares. The side length of each square is equal to the height of the semicircle at a given $x$, which is $2y = 2\sqrt{121 - x^2}$.

   Therefore, the area of the square at a particular $x$ is: $A(x) = \left( 2\sqrt{121 - x^2} \right)^2 = 4(121 - x^2) = 484 - 4x^2$

3. Volume by integration:
   To find the total volume, we integrate the area of the square cross-sections along the base of the semicircle from $x = -11$ to $x = 11$: $V = \int_{-11}^{11} A(x) \, dx = \int_{-11}^{11} (484 - 4x^2) \, dx$

4. Evaluate the integral:
   We can break the integral into two parts: $V = \int_{-11}^{11} 484 \, dx - \int_{-11}^{11} 4x^2 \, dx$ The first integral is straightforward: $\int_{-11}^{11} 484 \, dx = 484 \cdot (11 - (-11)) = 484 \cdot 22 = 10,648$ The second integral is: $\int_{-11}^{11} 4x^2 \, dx = 4 \cdot \left[ \frac{x^3}{3} \right]_{-11}^{11} = 4 \cdot \left( \frac{11^3}{3} - \frac{(-11)^3}{3} \right) = 4 \cdot \left( \frac{11^3}{3} - \frac{11^3}{3} \right) = 0$ Since the $x^2$ term is an even function, the integral over symmetric limits cancels out to zero.

   Therefore, the total volume is: $V = 10,648$

Thus, the volume of the solid is 10,648 cubic units.

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Would you like further details on any of these steps?

Here are five related questions for further exploration:

1. What is the formula for finding volumes of solids with different cross-sectional shapes?
2. How would the volume change if the base were a full circle rather than a semicircle?
3. Can this method be applied to cross-sections that are equilateral triangles instead of squares?
4. How does the shape of the base affect the volume of a solid?
5. What happens if the radius of the semicircular base is doubled?

Tip: When dealing with volume problems, always check if you can simplify the cross-sectional area function to make integration easier.