question find the net signed area between the...
question find 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. submit your answer as an exact value.
Answer
# Explanation:
## Step1: Rewrite the absolute - value function
For \(y = |x + 3|-4\), 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: Split the integral based on the break - point
We split the integral \(\int_{-9}^{3}(|x + 3|-4)dx=\int_{-9}^{-3}(-x - 7)dx+\int_{-3}^{3}(x - 1)dx\).
## Step3: Calculate the first integral
Using the power rule \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C(n\neq - 1)\), for \(\int_{-9}^{-3}(-x - 7)dx=\left[-\frac{x^{2}}{2}-7x\right]_{-9}^{-3}\).
\[
\begin{align*}
&(-\frac{(-3)^{2}}{2}-7\times(-3))-(-\frac{(-9)^{2}}{2}-7\times(-9))\\
=&(-\frac{9}{2}+21)-(-\frac{81}{2}+63)\\
=&(-\frac{9}{2}+21+\frac{81}{2}-63)\\
=&\frac{-9 + 81}{2}+21-63\\
=&\frac{72}{2}+21 - 63\\
=&36+21-63\\
=& - 6
\end{align*}
\]
## Step4: Calculate the second integral
For \(\int_{-3}^{3}(x - 1)dx=\left[\frac{x^{2}}{2}-x\right]_{-3}^{3}\).
\[
\begin{align*}
&(\frac{3^{2}}{2}-3)-(\frac{(-3)^{2}}{2}-(-3))\\
=&(\frac{9}{2}-3)-(\frac{9}{2}+3)\\
=&\frac{9}{2}-3-\frac{9}{2}-3\\
=& - 6
\end{align*}
\]
## Step5: Find the net - signed area
Add the results of the two integrals: \(\int_{-9}^{3}(|x + 3|-4)dx=-6+( - 6)=-12\).
# Answer:
\(-12\)