We are asked to find the volume of a solid that has a semicircular base with a radius of 13, and cross-sections perpendicular to the base and parallel to the diameter are squares.

Step-by-Step Process:

1. Understanding the base: The base of the solid is a semicircle, meaning we can define the semicircle in the $xy$-plane. The equation of a full circle with radius 13 centered at the origin is: $x^2 + y^2 = 13^2$ Since the base is a semicircle, we only consider the top half, where $y \geq 0$. This gives the equation for the semicircle as: $y = \sqrt{169 - x^2}$ The semicircle has a radius of 13, so the limits for $x$ are from $-13$ to $13$.

2. Cross-sectional area: We are told that the cross-sections perpendicular to the base and parallel to the diameter are squares. The side length of each square is determined by the width of the semicircle at a given $x$-value. This width is the distance from the top of the semicircle $y = \sqrt{169 - x^2}$ to the base of the semicircle, which is the $x$-axis ($y = 0$).

   Therefore, the side length $s(x)$ of each square at position $x$ is: $s(x) = 2 \cdot \sqrt{169 - x^2}$ since we are considering both the positive and negative parts of the semicircle.

3. Volume element: The volume of a small slice of the solid at position $x$ with thickness $dx$ is the area of the square cross-section times the thickness $dx$. The area of a square is $s(x)^2$, so the volume element is: $dV = s(x)^2 \, dx = \left( 2 \cdot \sqrt{169 - x^2} \right)^2 dx = 4(169 - x^2) dx$

4. Total volume: To find the total volume, we integrate the volume element over the entire base of the semicircle, which ranges from $x = -13$ to $x = 13$: $V = \int_{-13}^{13} 4(169 - x^2) \, dx$ This simplifies to: $V = 4 \int_{-13}^{13} (169 - x^2) \, dx$ We can now compute the integral.

5. Calculating the integral: Split the integral into two parts: $V = 4 \left[ \int_{-13}^{13} 169 \, dx - \int_{-13}^{13} x^2 \, dx \right]$ The first integral is straightforward: $\int_{-13}^{13} 169 \, dx = 169 \times (13 - (-13)) = 169 \times 26 = 4394$ The second integral is: $\int_{-13}^{13} x^2 \, dx$ This is a standard even function integral, and we know that: $\int_{-a}^{a} x^2 \, dx = \frac{2a^3}{3}$ So, in this case: $\int_{-13}^{13} x^2 \, dx = \frac{2(13)^3}{3} = \frac{2(2197)}{3} = \frac{4394}{3}$

6. Final volume: Now substitute these values back into the volume formula: $V = 4 \left( 4394 - \frac{4394}{3} \right) = 4 \left( \frac{3(4394) - 4394}{3} \right) = 4 \left( \frac{8788}{3} \right)$ Simplifying: $V = \frac{35152}{3}$

Thus, the exact volume of the solid is: $V = \frac{35152}{3} \text{ cubic units.}$

Would you like further details or have any questions?

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Here are 5 related questions:

1. How does the slicing method generalize to other cross-sectional shapes?
2. How would the volume change if the cross-sections were equilateral triangles instead of squares?
3. What is the geometric significance of integrating over a semicircle versus a full circle?
4. How does the radius of the semicircle affect the volume of the solid?
5. Can this method be extended to solids with different base shapes (e.g., ellipses)?

Tip: When dealing with cross-sectional volume problems, always clearly identify the shape and size of the cross-sections as functions of the position along the base.