Let the region mathrm{R} be the area enclosed by the function f(x)=e^{x}, the horizontal line y=3, 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 semi-circle with diameters extending through the region R, find the volume of the 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 (f(x) = e^x) and (y = 3) to determine the limits of integration: (e^x = 3) gives (x approx 1.099)Step:2 ：Integrate the area of each semi-circle along the x-axis: (V = int_{0}^{1.099} frac{1}{2} pi left(3 - e^xright)^2 dx approx 0.741)Step:3 ：The volume of the solid is approximately (boxed{0.741}) cubic units.

Answer

Step: 1 ：Find the intersection points of (f(x) = e^x) and (y = 3) to determine the limits of integration: (e^x = 3) gives (x approx 1.099)Step: 2 ：Integrate the area of each semi-circle along the x-axis: (V = int_{0}^{1.099} frac{1}{2} pi left(3 - e^xright)^2 dx approx 0.741)Step: 3 ：The volume of the solid is approximately (boxed{0.741}) cubic units.

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