A solid lies between planes perpendicular to the​ x-axis at xequals=minus−99 and xequals=99. The​ cross-sections perpendicular to the​ x-axis between these planes are squares whose bases run from the semicircle y equals negative StartRoot 81 minus x squared EndRooty=−81−x2 to the semicircle y equals StartRoot 81 minus x squared EndRooty=81−x2. Find the volume of the solid.

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.

A solid lies between planes perpendicular to the x-axis at x=-9 and x=9. The cross-sections perpendicular to the x-axis between these planes are squares whose bases run from the semicircle y = −√(81 − x^2) to the semicircle y = √(81 − x^2). Find the volume of the solid.

Learn how to find the volume of a solid with square cross-sections whose bases span between semicircles. This problem involves integral calculus and geometry, and it's suitable for college-level students. The detailed solution includes step-by-step integration and the application of geometric principles to calculate the volume of the solid.

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