let the region r be the area enclosed by the function f(x)=2e^x, the horizontal line y = 4, and the y…

Explanation:

Step1: Find the intersection point

Set $2e^{x}=4$, then $e^{x} = 2$, so $x=\ln2$.

Step2: Determine the height of the isosceles - right - triangle cross - section

The length of a leg of the isosceles right - triangle cross - section perpendicular to the $x$ - axis is $h = 4 - 2e^{x}$.

Step3: Find the area formula of the isosceles right - triangle cross - section

The area formula of an isosceles right - triangle is $A=\frac{1}{2}s^{2}$, where $s$ is the length of a leg. Here, $A(x)=\frac{1}{2}(4 - 2e^{x})^{2}=\frac{1}{2}(16-16e^{x}+4e^{2x}) = 8 - 8e^{x}+2e^{2x}$.

Step4: Calculate the volume using the integral

The volume $V$ of the solid with cross - sectional area $A(x)$ from $x = 0$ to $x=\ln2$ is given by the integral $V=\int_{0}^{\ln2}A(x)dx=\int_{0}^{\ln2}(8 - 8e^{x}+2e^{2x})dx$. Integrating term - by - term: $\int(8 - 8e^{x}+2e^{2x})dx=8x-8e^{x}+e^{2x}+C$. Evaluating the definite integral: [ \begin{align*} V&=\left[8x-8e^{x}+e^{2x}\right]_{0}^{\ln2}\ &=(8\ln2-8\times2 + 4)-(0 - 8+1)\ &=8\ln2-16 + 4+8 - 1\ &=8\ln2-5 \end{align*} ] Using a calculator, $8\ln2-5\approx8\times0.693147-5=5.545176 - 5=0.545$.

Answer:

$0.545$