A solid lies between planes perpendicular to the​ x-axis at xequals=0 and xequals=99. The​ cross-sections perpendicular to the axis on the interval 0less than or equals≤xless than or equals≤99 are squares with diagonals that run from the parabola y equals negative 2 StartRoot x EndRooty=−2x to the parabola y equals 2 StartRoot x EndRooty=2x. Find the volume of the solid.

To find the volume of the solid, we can use the method of cross-sections. Here's a step-by-step explanation of the approach.

### Step 1: Understanding the Problem
The solid lies between planes perpendicular to the \(x\)-axis at \(x = 0\) and \(x = 99\). The cross-sections of the solid perpendicular to the \(x\)-axis are squares, and the diagonals of these squares run between two parabolas:
- \(y = -2\sqrt{x}\)
- \(y = 2\sqrt{x}\)

### Step 2: Diagonal of the Square
The diagonal of each square is the vertical distance between the two parabolas, which is calculated as:
\[
\text{Diagonal} = 2\sqrt{x} - (-2\sqrt{x}) = 4\sqrt{x}
\]

### Step 3: Side Length of the Square
The diagonal \(d\) of a square is related to its side length \(s\) by the formula:
\[
d = s\sqrt{2}
\]
Thus, the side length \(s\) of each square is:
\[
s = \frac{d}{\sqrt{2}} = \frac{4\sqrt{x}}{\sqrt{2}} = 2\sqrt{2}\sqrt{x}
\]

### Step 4: Area of the Square
The area \(A(x)\) of each square cross-section is:
\[
A(x) = s^2 = \left(2\sqrt{2}\sqrt{x}\right)^2 = 8x
\]

### Step 5: Volume of the Solid
To find the volume of the solid, integrate the area of the cross-sections along the \(x\)-axis from \(x = 0\) to \(x = 99\):
\[
\text{Volume} = \int_0^{99} A(x) \, dx = \int_0^{99} 8x \, dx
\]

### Step 6: Perform the Integration
The integral of \(8x\) is:
\[
\int 8x \, dx = 8 \cdot \frac{x^2}{2} = 4x^2
\]
Thus, the volume is:
\[
\text{Volume} = \left[4x^2\right]_0^{99} = 4(99^2) - 4(0^2) = 4 \times 9801 = 39204
\]

### Final

Learn how to calculate the volume of a solid bounded by parabolic cross-sections, using squares as cross-sectional shapes. This problem involves integration techniques and geometric concepts. Suitable for advanced high school students.

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