find the total area between the graph of the ...

# Explanation: ## Step1: Find the x - intercept Set $f(x)=-x - 1=0$, then $x=-1$. ## Step2: Split the integral based on the x - intercept We split the interval $[-3,6]$ into $[-3,-1]$ and $[-1,6]$. The area $A=\int_{-3}^{-1}(-x - 1)dx+\int_{-1}^{6}(x + 1)dx$. ## Step3: Calculate the first integral $\int_{-3}^{-1}(-x - 1)dx=\left[-\frac{x^{2}}{2}-x\right]_{-3}^{-1}=\left(-\frac{(-1)^{2}}{2}-(-1)\right)-\left(-\frac{(-3)^{2}}{2}-(-3)\right)=\left(-\frac{1}{2}+1\right)-\left(-\frac{9}{2}+3\right)=\frac{1}{2}-\left(-\frac{3}{2}\right)=2$. ## Step4: Calculate the second integral $\int_{-1}^{6}(x + 1)dx=\left[\frac{x^{2}}{2}+x\right]_{-1}^{6}=\left(\frac{6^{2}}{2}+6\right)-\left(\frac{(-1)^{2}}{2}+(-1)\right)=(18 + 6)-\left(\frac{1}{2}-1\right)=24+\frac{1}{2}=\frac{49}{2}$. ## Step5: Sum the two areas $A = 2+\frac{49}{2}=\frac{4 + 49}{2}=\frac{53}{2}=26.5$. # Answer: $26.5$