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.

From Solvely APP
Source: https://solvelyapp.com/problems/8911/

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