let the region r be the area enclosed by the function f(x)=e^x and g(x)=4x + 1. if the region r is the base…

Explanation:

Step1: Find intersection points

Set $e^{x}=4x + 1$. Using a calculator, the intersection points are approximately $x = 0$ and $x\approx2.153$.

Step2: Determine base and height of rectangle

The base of the rectangle at a given $x$ is $b=e^{x}-(4x + 1)$. The height $h=\frac{1}{2}(e^{x}-(4x + 1))$.

Step3: Set up volume integral

The volume $V$ of the solid using the cross - sectional area formula $V=\int_{a}^{b}A(x)dx$, where $A(x)$ is the cross - sectional area. Here, $A(x)=b\times h=\frac{1}{2}(e^{x}-(4x + 1))^{2}$, and $a = 0$, $b\approx2.153$. So $V=\frac{1}{2}\int_{0}^{2.153}(e^{x}-4x - 1)^{2}dx$.

Step4: Expand integrand

$(e^{x}-4x - 1)^{2}=(e^{x})^{2}-2e^{x}(4x + 1)+(4x + 1)^{2}=e^{2x}-8xe^{x}-2e^{x}+16x^{2}+8x + 1$.

Step5: Integrate term - by - term

$\int e^{2x}dx=\frac{1}{2}e^{2x}$, $\int8xe^{x}dx=8(xe^{x}-e^{x})$ (using integration by parts), $\int2e^{x}dx = 2e^{x}$, $\int16x^{2}dx=\frac{16}{3}x^{3}$, $\int8xdx = 4x^{2}$, $\int1dx=x$.

Step6: Evaluate definite integral

$V=\frac{1}{2}\left[\frac{1}{2}e^{2x}-8(xe^{x}-e^{x})-2e^{x}+\frac{16}{3}x^{3}+4x^{2}+x\right]_{0}^{2.153}$. $V=\frac{1}{2}\left[\left(\frac{1}{2}e^{2\times2.153}-8(2.153e^{2.153}-e^{2.153})-2e^{2.153}+\frac{16}{3}(2.153)^{3}+4(2.153)^{2}+2.153\right)-\left(\frac{1}{2}-8(0 - 1)-2 + 0+0 + 0\right)\right]$. Using a calculator, $V\approx3.194$.

Answer:

$3.194$