The solid with a semicircular base of radius 11 whose cross sections perpendicular to the base and parallel to the diameter are squares. find volune


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.

The solid with a semicircular base of radius 11 whose cross sections perpendicular to the base and parallel to the diameter are squares. Find volume.

Find the volume of a solid with a semicircular base of radius 11 and square cross-sections perpendicular to the base. This problem explores the use of calculus to compute volume via cross-sectional area integration, covering key concepts in geometry and calculus. Suitable for advanced high school or college students.

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