let the region r be the area enclosed by the function f(x)=ln(x)-2 and g(x)=1/2x - 4. if the region r is the…

Explanation:

Step1: Find intersection points

Set $\ln(x)-2=\frac{1}{2}x - 4$. Using a calculator or software, we find the intersection points of $y = \ln(x)-2$ and $y=\frac{1}{2}x - 4$ are $x = 1$ and $x = 8$.

Step2: Determine diameter formula

The diameter $d$ of each semi - circle cross - section perpendicular to the $x$ - axis is $d=\left(\ln(x)-2\right)-\left(\frac{1}{2}x - 4\right)=\ln(x)-\frac{1}{2}x + 2$.

Step3: Find radius formula

The radius $r$ of each semi - circle is $r=\frac{1}{2}\left(\ln(x)-\frac{1}{2}x + 2\right)$.

Step4: Find area formula of semi - circle

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

Step5: Calculate volume using integral

The volume $V$ of the solid with cross - sectional area $A(x)$ from $x = a$ to $x = b$ is given by $V=\int_{a}^{b}A(x)dx$. Here, $a = 1$, $b = 8$, so $V=\int_{1}^{8}\frac{\pi}{8}\left(\ln(x)-\frac{1}{2}x + 2\right)^{2}dx$. Using a calculator to evaluate the integral $\int_{1}^{8}\frac{\pi}{8}\left(\ln(x)-\frac{1}{2}x + 2\right)^{2}dx\approx3.079$.

Answer:

$3.079$