use a graphing calculator to find the area between the functions accurate to the nearest thousandth.\nf(x) =…

Explanation:

Step1: Find intersection points

Use a graphing - calculator to find the intersection points of $y = e^{x}$ and $y=x + 5$. Let $e^{x}=x + 5$. By graphing, the intersection points are approximately $x_1=-2.317$ and $x_2 = 1.317$.

Step2: Set up the integral for the area

The area $A$ between two curves $y = f(x)$ and $y = g(x)$ is given by $A=\int_{a}^{b}|f(x)-g(x)|dx$. Here, for $-2.317\leq x\leq1.317$, $x + 5\geq e^{x}$, so $A=\int_{-2.317}^{1.317}((x + 5)-e^{x})dx$.

Step3: Integrate term - by - term

$\int_{-2.317}^{1.317}((x + 5)-e^{x})dx=\int_{-2.317}^{1.317}(x + 5)dx-\int_{-2.317}^{1.317}e^{x}dx$. We know that $\int(x + 5)dx=\frac{1}{2}x^{2}+5x+C$ and $\int e^{x}dx=e^{x}+C$. Evaluating $\int_{-2.317}^{1.317}(x + 5)dx=\left[\frac{1}{2}x^{2}+5x\right]{-2.317}^{1.317}=\left(\frac{1}{2}(1.317)^{2}+5(1.317)\right)-\left(\frac{1}{2}(-2.317)^{2}+5(-2.317)\right)$. Evaluating $\int{-2.317}^{1.317}e^{x}dx=\left[e^{x}\right]_{-2.317}^{1.317}=e^{1.317}-e^{-2.317}$. Then $A=\left(\frac{1}{2}(1.317)^{2}+5(1.317)\right)-\left(\frac{1}{2}(-2.317)^{2}+5(-2.317)\right)-(e^{1.317}-e^{-2.317})$.

Step4: Calculate the value

Using a calculator: $\frac{1}{2}(1.317)^{2}+5(1.317)=\frac{1}{2}\times1.734489+6.585 = 0.8672445+6.585=7.4522445$. $\frac{1}{2}(-2.317)^{2}+5(-2.317)=\frac{1}{2}\times5.368489-11.585 = 2.6842445-11.585=-8.9007555$. $e^{1.317}\approx3.739$, $e^{-2.317}\approx0.100$. $A=(7.4522445-(-8.9007555))-(3.739 - 0.100)$ $A=(7.4522445 + 8.9007555)-3.639$ $A = 16.353-3.639$ $A\approx12.714$.

Answer:

$12.714$