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.

Question

Accepted Answer

Step:1 ：Find the intersection points of (f(x) = 2sqrt{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 0.01667pi) cubic units.Step:5 ：(boxed{V approx 0.01667pi}) cubic units

Answer

Step: 1 ：Find the intersection points of (f(x) = 2sqrt{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 0.01667pi) cubic units.Step: 5 ：(boxed{V approx 0.01667pi}) cubic units

Answer

Popular Problems