question\nfind the net signed area between th...
question\nfind the net signed area between the graph of the function f(x) = |x + 3| - 4 and the x - axis over the interval -9,3, illustrated in the following image.\nsubmit your answer as an exact value.
Answer
# Explanation:
## Step1: Split the absolute - value function
The function \(y = f(x)=|x + 3|-4\) can be split into two cases. When \(x+3\geq0\) (i.e., \(x\geq - 3\)), \(y=(x + 3)-4=x - 1\); when \(x+3<0\) (i.e., \(x<-3\)), \(y=-(x + 3)-4=-x - 7\).
## Step2: Calculate the integral over sub - intervals
We split the interval \([-9,3]\) into two sub - intervals \([-9,-3]\) and \([-3,3]\).
The net - signed area \(A=\int_{-9}^{-3}(-x - 7)dx+\int_{-3}^{3}(x - 1)dx\).
First, calculate \(\int_{-9}^{-3}(-x - 7)dx\). Using the power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)\), we have \(\int(-x - 7)dx=-\frac{x^{2}}{2}-7x+C\).
Evaluating the definite integral: \(\left(-\frac{(-3)^{2}}{2}-7\times(-3)\right)-\left(-\frac{(-9)^{2}}{2}-7\times(-9)\right)\)
\[
\begin{align*}
&(-\frac{9}{2}+21)-(-\frac{81}{2}+63)\\
=&-\frac{9}{2}+21+\frac{81}{2}-63\\
=&\frac{-9 + 81}{2}+21-63\\
=&\frac{72}{2}+21 - 63\\
=&36+21-63\\
=& - 6
\end{align*}
\]
Second, calculate \(\int_{-3}^{3}(x - 1)dx\). Using the power rule, \(\int(x - 1)dx=\frac{x^{2}}{2}-x+C\).
Evaluating the definite integral: \(\left(\frac{3^{2}}{2}-3\right)-\left(\frac{(-3)^{2}}{2}-(-3)\right)\)
\[
\begin{align*}
&(\frac{9}{2}-3)-(\frac{9}{2}+3)\\
=&\frac{9}{2}-3-\frac{9}{2}-3\\
=& - 6
\end{align*}
\]
## Step3: Sum the results of the definite integrals
\(A=-6+( - 6)=-12\)
# Answer:
\(-12\)