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.

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Accepted Answer

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 2sqrt{2})Step:2 ：Integrate the area of the square cross-sections along the x-axis: (int_{-2sqrt{2}}^{2sqrt{2}} (8 - x^2)^2 dx = frac{2048sqrt{2}}{15})Step:3 ：boxed{text{Final Answer: The volume of the solid is approximately } 193.087 text{ cubic units}}

Answer

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 2sqrt{2})Step: 2 ：Integrate the area of the square cross-sections along the x-axis: (int_{-2sqrt{2}}^{2sqrt{2}} (8 - x^2)^2 dx = frac{2048sqrt{2}}{15})Step: 3 ：boxed{text{Final Answer: The volume of the solid is approximately } 193.087 text{ cubic units}}

Answer

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