let the region r be the area enclosed by the function f(x)=3ln(x) and g(x)=2x - 3. if the region r is the…

let the region r be the area enclosed by the function f(x)=3ln(x) and g(x)=2x - 3. if the region 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.

let the region r be the area enclosed by the function f(x)=3ln(x) and g(x)=2x - 3. if the region 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.

Answer

Explanation:

Step1: Find intersection points

Set $3\ln(x)=2x - 3$. Using a calculator, the intersection points are $x = 1$ and $x\approx3.146$.

Step2: Determine diameter formula

The diameter $d$ of each semi - circle cross - section is $d=(2x - 3)-3\ln(x)$.

Step3: Find radius formula

The radius $r=\frac{(2x - 3)-3\ln(x)}{2}$.

Step4: Find area formula of semi - circle

The area of a semi - circle $A=\frac{1}{2}\pi r^{2}=\frac{\pi}{8}((2x - 3)-3\ln(x))^{2}$.

Step5: Calculate volume using integral

The volume $V=\int_{1}^{3.146}\frac{\pi}{8}((2x - 3)-3\ln(x))^{2}dx$. Using a calculator to evaluate the integral: $V\approx1.173$.

Answer:

$1.173$