question find the total area between the graph of the function f(x)=-x - 2 graphed below, and the x - axis…

Explanation:

Step1: Find the x - intercept

Set $f(x)=0$, so $-x - 2=0$, then $x=-2$.

Step2: Split the integral based on the x - intercept

The area $A=\int_{-6}^{-2}(-x - 2)dx+\left|\int_{-2}^{4}(-x - 2)dx\right|$.

Step3: Calculate $\int_{-6}^{-2}(-x - 2)dx$

Using the power - rule $\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)$, we have $\int(-x - 2)dx=-\frac{x^{2}}{2}-2x+C$. Evaluating $\left(-\frac{x^{2}}{2}-2x\right)\big|_{-6}^{-2}=(-\frac{(-2)^{2}}{2}-2\times(-2))-(-\frac{(-6)^{2}}{2}-2\times(-6))=( - 2 + 4)-(-18 + 12)=2-(-6)=8$.

Step4: Calculate $\int_{-2}^{4}(-x - 2)dx$

Evaluating $\left(-\frac{x^{2}}{2}-2x\right)\big|{-2}^{4}=(-\frac{4^{2}}{2}-2\times4)-(-\frac{(-2)^{2}}{2}-2\times(-2))=(-8 - 8)-(-2 + 4)=-16 - 2=-18$. Take the absolute value, $\left|\int{-2}^{4}(-x - 2)dx\right| = 18$.

Step5: Sum the two areas

$A=8 + 18=26$.

Answer:

26