question find the total area between the graph of the function f(x)=−x−1, graphed below, and the x - axis…

Explanation:

Step1: Find the x - intercept

Set $f(x)=-x - 1=0$, then $x=-1$.

Step2: Split the integral based on the x - intercept

The area $A=\int_{-6}^{-1}(-x - 1)dx+\int_{-1}^{3}-(-x - 1)dx$.

Step3: Calculate the first integral

$\int_{-6}^{-1}(-x - 1)dx=\left[-\frac{1}{2}x^{2}-x\right]_{-6}^{-1}=\left(-\frac{1}{2}\times(-1)^{2}-(-1)\right)-\left(-\frac{1}{2}\times(-6)^{2}-(-6)\right)=\left(-\frac{1}{2}+1\right)-\left(-18 + 6\right)=\frac{1}{2}+12=\frac{25}{2}$.

Step4: Calculate the second integral

$\int_{-1}^{3}-(-x - 1)dx=\int_{-1}^{3}(x + 1)dx=\left[\frac{1}{2}x^{2}+x\right]_{-1}^{3}=\left(\frac{1}{2}\times3^{2}+3\right)-\left(\frac{1}{2}\times(-1)^{2}+(-1)\right)=\left(\frac{9}{2}+3\right)-\left(\frac{1}{2}-1\right)=\frac{15}{2}+\frac{1}{2}=8$.

Step5: Sum up the two areas

$A=\frac{25}{2}+8=\frac{25 + 16}{2}=\frac{41}{2}=20.5$.

Answer:

$20.5$