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