find the total area between the graph of the ...
find the total area between the graph of the function f(x) = x + 1 and the x - axis over the interval -4,6. 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 $A=\int_{-4}^{-1}-(x + 1)dx+\int_{-1}^{6}(x + 1)dx$.
## Step3: Integrate $-(x + 1)$
$\int_{-4}^{-1}-(x + 1)dx=-\left[\frac{x^{2}}{2}+x\right]_{-4}^{-1}=-\left(\left(\frac{(-1)^{2}}{2}-1\right)-\left(\frac{(-4)^{2}}{2}-4\right)\right)=-\left(\frac{1}{2}-1 - 8 + 4\right)=-\left(\frac{1}{2}-5\right)=\frac{9}{2}$.
## Step4: Integrate $(x + 1)$
$\int_{-1}^{6}(x + 1)dx=\left[\frac{x^{2}}{2}+x\right]_{-1}^{6}=\left(\frac{6^{2}}{2}+6\right)-\left(\frac{(-1)^{2}}{2}-1\right)=\left(18 + 6\right)-\left(\frac{1}{2}-1\right)=24+\frac{1}{2}=\frac{49}{2}$.
## Step5: Calculate the total area
$A=\frac{9}{2}+\frac{49}{2}=\frac{9 + 49}{2}=29$.
# Answer:
$29$