Problem
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.
Solution
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: \(V = \int_{0}^{1.099} \frac{1}{2} \pi \left(3 - e^x\right)^2 dx \approx 0.741\)
Step 3 ：The volume of the solid is approximately \(\boxed{0.741}\) cubic units.
