Let the region $\mathrm{R}$ be the area enclosed by the function $f(x)=x^{2}-2$, the horizontal line $y=6$, and the $y$-axis. If the region $\mathrm{R}$ is the base of a solid such that each cross section perpendicular to the $x$-axis is a square, find the volume of પ̇he solid. You may use a calculator and round to the nearest thousandth.
Answer
\boxed{\text{Final Answer: The volume of the solid is approximately } 193.087 \text{ cubic units}}
Step 1 ：Find the intersection points of the function \(f(x) = x^2 - 2\) and the horizontal line \(y = 6\) to determine the bounds of integration: \(x = \pm 2\sqrt{2}\)
Step 2 ：Integrate the area of the square cross-sections along the x-axis: \(\int_{-2\sqrt{2}}^{2\sqrt{2}} (8 - x^2)^2 dx = \frac{2048\sqrt{2}}{15}\)
Step 3 ：\boxed{\text{Final Answer: The volume of the solid is approximately } 193.087 \text{ cubic units}}
