Let the region $\mathrm{R}$ be the area enclosed by the function $f(x)=2 \sqrt{x}$ and $g(x)=2 x$. 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 $\mathrm{R}$, find the volume of the solid. You may use a calculator and round to the nearest thousandth.
Answer
\(\boxed{V \approx 0.01667\pi}\) cubic units
Step 1 ：Find the intersection points of \(f(x) = 2\sqrt{x}\) and \(g(x) = 2x\) to determine the limits of integration.
Step 2 ：Set up the integral for the volume of the solid using the formula for the area of a semi-circle: \(A = \frac{1}{2}\pi r^2\), where \(r = \sqrt{x} - x\).
Step 3 ：Calculate the volume using the integral: \(V = \int_0^1 \frac{1}{2}\pi (\sqrt{x} - x)^2 dx\).
Step 4 ：Evaluate the integral to find the volume: \(V \approx 0.01667\pi\) cubic units.
Step 5 ：\(\boxed{V \approx 0.01667\pi}\) cubic units
