The problem asks to find the volume of a solid whose base is a semicircle described by the equation $y = \sqrt{25 - x^2}$, with the cross sections perpendicular to the x-axis being squares. The interval for $x$ is from $-5$ to $5$.

Step 1: Understand the Geometry of the Problem

• The base is a semicircle, and the equation $y = \sqrt{25 - x^2}$ describes the upper half of a circle with radius 5, centered at the origin. The value of $y$ at any point on the x-axis is the height of the semicircle.

• The cross sections perpendicular to the x-axis are squares. This means that at each point $x$, the side length of the square is equal to $2y$, which is the vertical distance from the x-axis to the curve.

Step 2: Formula for the Volume

The volume $V$ of the solid can be computed by integrating the area of the cross-sectional square along the x-axis:

$V = \int_{-5}^{5} A(x) \, dx$ where $A(x)$ is the area of the square at each point. Since the side length of each square is $2y = 2\sqrt{25 - x^2}$, the area of the square is:

$A(x) = \left( 2\sqrt{25 - x^2} \right)^2 = 4(25 - x^2)$

Thus, the volume integral becomes:

$V = \int_{-5}^{5} 4(25 - x^2) \, dx$

Step 3: Evaluate the Integral

We can now solve this definite integral:

$V = 4 \int_{-5}^{5} (25 - x^2) \, dx$

We can separate this into two integrals:

$V = 4 \left( \int_{-5}^{5} 25 \, dx - \int_{-5}^{5} x^2 \, dx \right)$

1. The first integral: $\int_{-5}^{5} 25 \, dx = 25 \cdot (5 - (-5)) = 25 \cdot 10 = 250$

2. The second integral: $\int_{-5}^{5} x^2 \, dx = 2 \int_{0}^{5} x^2 \, dx = 2 \cdot \left[ \frac{x^3}{3} \right]_{0}^{5} = 2 \cdot \left( \frac{125}{3} \right) = \frac{250}{3}$

Step 4: Combine the Results

Now, substituting the results back into the volume equation:

$V = 4 \left( 250 - \frac{250}{3} \right)$

Simplifying:

$V = 4 \cdot \left( \frac{750}{3} - \frac{250}{3} \right) = 4 \cdot \frac{500}{3} = \frac{2000}{3}$

Final Answer:

The volume of the solid is $\frac{2000}{3}$ cubic units.

Would you like further explanation on any part of the solution? Feel free to ask. Here are some related questions to explore:

1. How would the volume change if the cross sections were triangles instead of squares?
2. How can we compute the volume of solids with different geometric shapes for cross sections?
3. What other types of integrals are used in geometry problems like this?
4. How do you determine the area of other shapes of cross sections (e.g., rectangles, circles)?
5. What does the symmetry of this problem tell us about solving the integral efficiently?

Tip: When dealing with symmetrical regions (like semicircles or full circles), it's often helpful to split the integral into two parts to simplify calculations, especially when limits are symmetric about the origin.