question find the total area between the grap...
question find the total area between the graph of the function f(x)=−x−1, graphed below, and the x - axis over the interval −6,3. provide your answer below. a =
Answer
# 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:
$20.5$